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Lesson 1: Three-Dimensional Cartesian Coordinates and Vector Operations — Picture questions (exam style)

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1.
Plane: 3x + 2y + -2z = 5; the line passes through P(3, 2, 1) with direction (1, 0, 1). Which option is correct?
Plane: 3x + 2y + -2z = 5; the line passes through P(3, 2, 1) with direction (1, 0, 1). Which option is correct?
angle between line and plane ≈ 4.9°, distance of P from the plane = 6
angle between line and plane ≈ 9.9°, distance of P from the plane = 1.46
angle between line and plane ≈ 4.9°, distance of P from the plane = 1.46
angle between line and plane ≈ 4.9°, distance of P from the plane = 3.88
2.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
14.97
7.483
3.5
8.261
3.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
46.23
-11
-77
77
4.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
19
7.228
20.33
14.46
5.
Find the length of AB.
Find the length of AB.
136
11.66
7.211
16
6.
Find angle BAC in degrees.
Find angle BAC in degrees.
122.3 °
28.84 °
32.32 °
57.68 °
7.
Find angle BAC in degrees.
Find angle BAC in degrees.
70.53 °
109.5 °
19.47 °
35.26 °
8.
Find the length of AB.
Find the length of AB.
7
13
81
9
9.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
21.21
19.21
38.42
9
10.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
14
53
15
-53
11.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
40.25
150
-150
-30
12.
Find the length of AB.
Find the length of AB.
10.05
101
5.745
15
13.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-6
14.49
6
2
14.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-56
56
64.51
-10
15.
Find the length of AB.
Find the length of AB.
10
9.274
6.164
38
16.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
72
6.928
1.504
0.973
17.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
3
4.583
0.1474
0.1
18.
Find angle BAC in degrees.
Find angle BAC in degrees.
37.59 °
52.41 °
18.79 °
142.4 °
19.
Find the length of AB.
Find the length of AB.
7.348
7.071
12
50
20.
Find angle BAC in degrees.
Find angle BAC in degrees.
155.9 °
24.12 °
12.06 °
65.88 °
21.
Find the length of AB.
Find the length of AB.
8.307
69
7.28
13
22.
Find angle BAC in degrees.
Find angle BAC in degrees.
74.58 °
105.4 °
52.71 °
15.42 °
23.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-22
6
-6
41.83
24.
Find the length of AB.
Find the length of AB.
11
5
45
6.708
25.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
31.4
29
15.7
32.98
26.
Find angle BAC in degrees.
Find angle BAC in degrees.
70.57 °
19.43 °
35.29 °
109.4 °
27.
Find angle BAC in degrees.
Find angle BAC in degrees.
116.7 °
58.34 °
63.32 °
26.68 °
28.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
28
2
0.9333
1.484
29.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
24
69.57
34.79
42.26
30.
Find the length of AB.
Find the length of AB.
3.606
5
8.544
13
31.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-3
-1
1
48.83
32.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
45
1.225
0.75
3.742
33.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
9.028
2
18.06
9.247
34.
Find the length of AB.
Find the length of AB.
9.11
7.141
51
11
35.
Find angle BAC in degrees.
Find angle BAC in degrees.
87.79 °
46.11 °
92.21 °
2.212 °
36.
Find angle BAC in degrees.
Find angle BAC in degrees.
14.09 °
151.8 °
28.19 °
61.81 °
37.
Find angle BAC in degrees.
Find angle BAC in degrees.
168 °
12 °
78 °
5.999 °
38.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
29
38.79
14
-29
39.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
57.25
0
-3
3
40.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.07692
6
0.1275
5.099
41.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
54
-54
12.73
-1
42.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
83
5
2.515
3.657
43.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
6
27
-36
-6
44.
Find the length of AB.
Find the length of AB.
50
8
7.071
9.695
45.
Find the length of AB.
Find the length of AB.
6.633
10
11.54
44
46.
Find angle BAC in degrees.
Find angle BAC in degrees.
35.56 °
108.9 °
18.88 °
71.12 °
47.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
13.86
18.33
27.71
12
48.
Find the length of AB.
Find the length of AB.
6.164
38
7.071
8
49.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
6
5.099
0.07692
0.128
50.
Find angle BAC in degrees.
Find angle BAC in degrees.
110.4 °
34.79 °
20.41 °
69.59 °
51.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
32.65
-3
28
-28
52.
Find angle BAC in degrees.
Find angle BAC in degrees.
42 °
21 °
138 °
48 °
53.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
25.17
26.41
50.34
8
54.
Find the length of AB.
Find the length of AB.
2
4
10.2
3.48
55.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
36.88
73.32
4
36.66
56.
Find angle BAC in degrees.
Find angle BAC in degrees.
35.43 °
54.57 °
17.72 °
144.6 °
57.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
45.87
-19
9
19
58.
Find the length of AB.
Find the length of AB.
7.874
6.481
62
12
59.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-10
41.11
-26
26
60.
Find angle BAC in degrees.
Find angle BAC in degrees.
13.15 °
103.1 °
38.43 °
76.85 °
61.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
2.857
5.745
80
3.288
62.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
34
32.94
65.88
47.34
63.
Find angle BAC in degrees.
Find angle BAC in degrees.
26.58 °
58.29 °
63.42 °
116.6 °
64.
Find the length of AB.
Find the length of AB.
41
12.69
6.403
9
65.
Find angle BAC in degrees.
Find angle BAC in degrees.
99.69 °
9.685 °
40.16 °
80.31 °
66.
Find angle BAC in degrees.
Find angle BAC in degrees.
24.27 °
32.87 °
114.3 °
65.73 °
67.
Find the length of AB.
Find the length of AB.
4.583
11
7.28
53
68.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
7.071
1.974
77
3.013
69.
Find the length of AB.
Find the length of AB.
84
4.899
9.165
14
70.
Find angle BAC in degrees.
Find angle BAC in degrees.
123.6 °
28.19 °
56.38 °
33.62 °
71.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
26
32.7
16.35
30.71
72.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
12
50.7
28.04
25.35
73.
Find the length of AB.
Find the length of AB.
5.099
11.92
8
26
74.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
8.078
7.5
15
3
75.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-90
90
-45
43.59
76.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.6438
5
40
0.3922
77.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-94
20.1
-6
94
78.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-53
-3
37.54
53
79.
Find the length of AB.
Find the length of AB.
5.831
16
9.274
86
80.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
8.124
1.294
176
1.922
81.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
57.45
-10
10
6
82.
Find the length of AB.
Find the length of AB.
8.602
8.124
14
74
83.
Find the length of AB.
Find the length of AB.
46
10
7.616
6.782
84.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
25
-8
41.23
8
85.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
19.47
4.5
10.72
9.734
86.
Find angle BAC in degrees.
Find angle BAC in degrees.
148.8 °
15.6 °
31.2 °
58.8 °
87.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
44.93
-25
-29
29
88.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-71
71
98.51
-3
89.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
35.9
-6
6
4
90.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-19
19
33
6.708
91.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
5.196
200
4.642
2.778
92.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
1.974
2.933
5.099
77
93.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
3
22.05
22.25
44.09
94.
Find the length of AB.
Find the length of AB.
8.602
74
12
10.49
95.
Find angle BAC in degrees.
Find angle BAC in degrees.
5.558 °
47.78 °
95.56 °
84.44 °
96.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.8387
3.162
1.412
52
97.
Find the length of AB.
Find the length of AB.
21
4.583
7
6.403
98.
Find angle BAC in degrees.
Find angle BAC in degrees.
100.5 °
50.26 °
79.48 °
10.52 °
99.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
3.016
233
8.124
2.219
100.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-56
43.15
56
8
101.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
90.75
-11
-35
35
102.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
30.51
22.6
20.5
45.2
103.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
16.16
5
16.91
32.31
104.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
92
6.481
2.875
3.707
105.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
17
45.3
41.99
83.98
106.
Find the length of AB.
Find the length of AB.
7.55
57
11
8.307
107.
Find the length of AB.
Find the length of AB.
9.487
14
4.243
90
108.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
6
33.61
67.22
34.14
109.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
2.5
15.33
8.062
7.665
110.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-21
21
24.74
19
111.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-16
90
48.56
-90
112.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
2.373
178
5.831
2.854
113.
Find angle BAC in degrees.
Find angle BAC in degrees.
93.8 °
46.9 °
3.797 °
86.2 °
114.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-15
15
53.24
12
115.
Find angle BAC in degrees.
Find angle BAC in degrees.
28.71 °
32.58 °
57.42 °
122.6 °
116.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
3.742
28
28.06
1.871
117.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
19.34
30
38.68
35.69
118.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
20.45
35.67
17.83
10
119.
Find angle BAC in degrees.
Find angle BAC in degrees.
156.6 °
45 °
0 °
90 °
120.
Find angle BAC in degrees.
Find angle BAC in degrees.
33.61 °
56.39 °
16.81 °
146.4 °
121.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.1037
2
7.071
0.06667
122.
Find the length of AB.
Find the length of AB.
12.69
161
21
3
123.
Find the length of AB.
Find the length of AB.
8.775
6.708
77
13
124.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
13
-13
-6
19.42
125.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
10
90
-90
69.32
126.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
187
4.16
2.791
7.681
127.
Find the length of AB.
Find the length of AB.
69
14.45
13
8.307
128.
Find the length of AB.
Find the length of AB.
8.062
7
37
6.083
129.
Find the length of AB.
Find the length of AB.
22
7.071
8
4.69
130.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-54
27.93
-4
54
131.
Find angle BAC in degrees.
Find angle BAC in degrees.
48.99 °
20.51 °
139 °
41.01 °
132.
Find the length of AB.
Find the length of AB.
21
7
6.708
4.583
133.
Find angle BAC in degrees.
Find angle BAC in degrees.
40.7 °
49.3 °
24.65 °
130.7 °
134.
Find the length of AB.
Find the length of AB.
9
11
7.55
57
135.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
19.5
21.11
16.16
8.078
136.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
31
40.3
80.6
50.84
137.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
1.411
0.88
6.164
132
138.
Find the length of AB.
Find the length of AB.
11.05
4
3.162
10
139.
Find the length of AB.
Find the length of AB.
50
10
6.481
7.071
140.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-24
28.28
24
6
141.
Find the length of AB.
Find the length of AB.
66
8.124
10
7.348
142.
Find the length of AB.
Find the length of AB.
9.487
14
6
3.742
143.
Find angle BAC in degrees.
Find angle BAC in degrees.
68.8 °
42.39 °
137.6 °
47.61 °
144.
Find angle BAC in degrees.
Find angle BAC in degrees.
54.47 °
71.07 °
108.9 °
18.93 °
145.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
37.99
75.97
39
54.44
146.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
49.5
0
25
1
147.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-23
40.02
23
12.08
148.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
12
4
12.65
8
149.
Find the length of AB.
Find the length of AB.
98
16
9.899
7.616
150.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-90
90
-32
76.16
151.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-11
11
28.14
12
152.
Find angle BAC in degrees.
Find angle BAC in degrees.
21.14 °
68.86 °
55.57 °
111.1 °
153.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
12.73
6.364
32.5
33.12
154.
Find angle BAC in degrees.
Find angle BAC in degrees.
117.5 °
62.52 °
27.48 °
31.26 °
155.
Find angle BAC in degrees.
Find angle BAC in degrees.
26.14 °
153.9 °
63.86 °
13.07 °
156.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-6
40.99
10.44
6
157.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.4989
4.359
34
0.2982
158.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
3.742
67
1.426
2.232
159.
Find angle BAC in degrees.
Find angle BAC in degrees.
65.33 °
114.7 °
24.67 °
32.66 °
160.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
85.85
27
-27
-15
161.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
43.96
45.99
13.5
87.93
162.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
45
5.831
0.7106
0.4412
163.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
4.472
40
0.6349
0.8412
164.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
26.05
52.1
27.25
8
165.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
7.071
3.536
26.24
26
166.
Find angle BAC in degrees.
Find angle BAC in degrees.
51.18 °
102.4 °
77.64 °
12.36 °
167.
Find angle BAC in degrees.
Find angle BAC in degrees.
63.6 °
116.4 °
31.8 °
26.4 °
168.
Find angle BAC in degrees.
Find angle BAC in degrees.
61.75 °
28.25 °
118.3 °
30.87 °
169.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
30
6
14.07
-6
170.
Find angle BAC in degrees.
Find angle BAC in degrees.
5.333 °
10.67 °
169.3 °
79.33 °
171.
Find angle BAC in degrees.
Find angle BAC in degrees.
8.92 °
162.2 °
72.16 °
17.84 °
172.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
18.5
28.65
23.4
14.33
173.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-2
-38
50.58
38
174.
Find angle BAC in degrees.
Find angle BAC in degrees.
33.06 °
56.94 °
16.53 °
146.9 °
175.
Find the length of AB.
Find the length of AB.
101
10.05
15
7.28
176.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-11
16
-16
43.75
177.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
4
6.403
0.1185
0.07407
178.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
40
44.02
36.77
18.38
179.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-23
20
14.46
23
180.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.25
2.236
9
0.4049
181.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
26.42
52.84
2
26.5
182.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
2.484
231
3.319
4.243
183.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-1
39.76
-83
83
184.
Find the length of AB.
Find the length of AB.
2.236
3
7
5
185.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.05
6.403
5
0.07762
186.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
20
3.742
1.006
0.6061
187.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
62.64
-9
9
-2
188.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
81.82
-70
70
12
189.
Find the length of AB.
Find the length of AB.
7.483
104
10.2
16
190.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
5.477
109
4.909
3.114
191.
Find angle BAC in degrees.
Find angle BAC in degrees.
38.59 °
77.18 °
12.82 °
102.8 °
192.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
3.317
55
1.774
3.032
193.
Find the length of AB.
Find the length of AB.
9.539
13
7.681
91
194.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
3.529
2.151
157
4.583
195.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
7
25.42
72
-72
196.
Find the length of AB.
Find the length of AB.
8.124
6.164
10
38
197.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
35
0.5833
7.071
0.8329
198.
Find angle BAC in degrees.
Find angle BAC in degrees.
67.62 °
22.38 °
33.81 °
112.4 °
199.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
31.12
5
30.72
61.43
200.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
5.099
0.8462
1.193
66
201.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
72
2.057
5.385
2.947
202.
Find the length of AB.
Find the length of AB.
20
138
4.69
11.75
203.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
9.682
18.06
3.5
9.028
204.
Find angle BAC in degrees.
Find angle BAC in degrees.
120.1 °
29.95 °
59.89 °
30.11 °
205.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
24.92
11
44.72
22.36
206.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
46.95
16.37
44
32.74
207.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
50
1.385
0.8333
4.123
208.
Find the length of AB.
Find the length of AB.
144
12
6.633
20
209.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
1.746
48
6
1.143
210.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
28.53
2
14.27
14.4
211.
Find the length of AB.
Find the length of AB.
8.062
9
12.21
65
212.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
45
43.43
86.86
62.54
213.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
14.9
10.22
7
7.45
214.
Find the length of AB.
Find the length of AB.
7
27
6.557
5.196
215.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
19.5
23.48
46.96
30.52
216.
Find the length of AB.
Find the length of AB.
6.708
45
9
11
217.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
0
26
-26
39.45
218.
Find the length of AB.
Find the length of AB.
8.062
13
6.083
65
219.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
64
4.69
1.333
1.651
220.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
38.42
20.12
6
19.21
221.
Find angle BAC in degrees.
Find angle BAC in degrees.
123.2 °
33.21 °
56.79 °
28.39 °
222.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
147
5.196
1.4
2.333
223.
Find the length of AB.
Find the length of AB.
10
6.782
46
4.243
224.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
33.23
18
16.61
24.49
225.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
7.681
0.6
0.9931
12
226.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
1.609
111
2.72
6.403
227.
Find the length of AB.
Find the length of AB.
125
11.18
17
6.083
228.
Find the length of AB.
Find the length of AB.
10
5.477
46
6.782
229.
Find the length of AB.
Find the length of AB.
11
49
10.63
7
230.
Find angle BAC in degrees.
Find angle BAC in degrees.
121.7 °
31.65 °
29.17 °
58.35 °
231.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
12
33.97
31.78
63.57
232.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
32
7.348
0.5161
0.784
233.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
30.89
15.44
13.5
20.51
234.
Find angle BAC in degrees.
Find angle BAC in degrees.
137.3 °
21.36 °
42.72 °
47.28 °
235.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
28.43
14.21
11
17.97
236.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
92
3
1.853
1.438
237.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
45.5
48.79
35.24
17.62
238.
Find angle BAC in degrees.
Find angle BAC in degrees.
106.9 °
73.14 °
16.86 °
53.43 °
239.
Find angle BAC in degrees.
Find angle BAC in degrees.
62.07 °
13.96 °
152.1 °
27.93 °
240.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
11.51
9.5
13
6.5
241.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
31.11
16
-16
-9
242.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-3
1
115
0
243.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
2
18.04
17.93
35.86
244.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-24
47.22
-7
24
245.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
6.481
2.616
1.652
38
246.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
30.81
31.07
61.62
4
247.
Find angle BAC in degrees.
Find angle BAC in degrees.
64.36 °
25.64 °
12.82 °
154.4 °
248.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
30
1.448
1.111
5.385
249.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
4.583
3.28
1.909
42
250.
Find angle BAC in degrees.
Find angle BAC in degrees.
41.05 °
24.47 °
131.1 °
48.95 °
251.
Find angle BAC in degrees.
Find angle BAC in degrees.
20.48 °
40.95 °
49.05 °
139 °
252.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-1
11
-11
13.15
253.
Find angle BAC in degrees.
Find angle BAC in degrees.
122.6 °
28.71 °
57.43 °
32.57 °
254.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
40.47
-98
-35
98
255.
Find the length of AB.
Find the length of AB.
93
15
6.708
9.644
256.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
17.72
21
6
-6
257.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
2.995
106
6
1.738
258.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
62.86
58.88
22
117.8
259.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.9341
0.6338
45
5.385
260.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
12.5
30.31
55.23
27.61
261.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-16
16
18.14
9
262.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
32.45
17.87
7.5
16.22
263.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
58.74
39.5
43.48
86.96
264.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
38.64
27.1
19.32
19
265.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
100
5
1.042
1.648
266.
Find angle BAC in degrees.
Find angle BAC in degrees.
91.95 °
88.05 °
44.02 °
1.954 °
267.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
120
2.143
3
5.099
268.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
39
27
-4
-39
269.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
41.02
76.37
15
38.18
270.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
66
0.7333
1.254
4.123
271.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
16.02
10.61
12
21.21
272.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
41.32
10.5
79.94
39.97
273.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
13.5
12.83
18.62
25.65
274.
Find the length of AB.
Find the length of AB.
2.449
114
18
10.68
275.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
59.67
72
-72
-34
276.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
47.57
44.68
42
22.34
277.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-7
22.91
7
-1
278.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
4
57.5
-3
3
279.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
3.996
6.481
3.1
279
280.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
16
-5
-16
28.28
281.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
37.42
18.71
42.5
46.44
282.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.9623
0.72
6.708
54
283.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.0303
0.04192
4.359
2
284.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
63.16
56
29.21
58.43
285.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
26.5
52
-52
-10
286.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.2277
4.123
23
0.3676
287.
Find the length of AB.
Find the length of AB.
50
7.071
8
9.487
288.
Find the length of AB.
Find the length of AB.
15
9.539
9.95
99
289.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.5122
6.481
0.3051
36
290.
Find the length of AB.
Find the length of AB.
7.348
10
10.86
54
291.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
32.76
65.52
4.5
33.07
292.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
84.43
-50
50
-2
293.
Find the length of AB.
Find the length of AB.
13
81
5.745
9
294.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
8
21
-8
28
295.
Find the length of AB.
Find the length of AB.
9.274
16
86
4.243
296.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-2
85
39.57
-85
297.
Find the area of triangle ABC (use the vector product).
Find the area of triangle ABC (use the vector product).
12.5
6.225
12.45
13.96
298.
Find angle BAC in degrees.
Find angle BAC in degrees.
70.71 °
19.29 °
35.36 °
109.3 °
299.
Find the perpendicular distance from the origin to the plane through A, B, C.
Find the perpendicular distance from the origin to the plane through A, B, C.
0.8274
78
0.4875
6.481
300.
Find the scalar product AB · AC.
Find the scalar product AB · AC.
-7
8
9.274
7
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