1.1.2Dynamics
Calculations with Vectors
5 min read 5 parts Lesson 2 of 22
Section 1 of 5Adding Vectors in a Straight Line
Combining vectors is different from combining scalars because the directions must be taken into account.
Same direction
- When two vectors point the same way, add their sizes: 5 N and 3 N to the right give a resultant of 8 N to the right.
Opposite directions
- When vectors point in opposite ways, subtract the smaller from the larger and keep the direction of the larger one.
Resultant
- The resultant is the single vector that has exactly the same effect as all the original vectors acting together.
Sign method
- Calling right positive and left negative lets you simply add the signed numbers to get the resultant and its direction.
Section 2 of 5Vectors at Right Angles
Vectors at Right Angles
Tip-to-tail
- Draw the first vector, start the second where the first ends, and the resultant joins the very start to the very finish.
Pythagoras
- For two perpendicular vectors, the size of the resultant is found from R² = a² + b², so R = √(a² + b²).
Angle
- The direction is found from tan θ = opposite ÷ adjacent, using the two original vector sizes, then θ = tan⁻¹ of that ratio.
Bearing
- A bearing is an angle measured clockwise from north written with three digits, so due east is 090° and south-west is 225°.
| Situation | Method | Example |
|---|---|---|
| Same line, same way | Add | 4 m + 6 m = 10 m |
| Same line, opposite | Subtract | 9 N − 5 N = 4 N |
| Right angles | Pythagoras + tan θ | 3 km N + 4 km E = 5 km |
Section 3 of 5Applying the Method to Motion and Forces
1
Sketch the vectors tip-to-tail so you can see the right-angled triangle that the problem creates.
2
Label the two perpendicular sides with their sizes and units, and mark the unknown resultant as the hypotenuse.
3
Calculate the resultant size using Pythagoras and round sensibly, keeping the unit with the answer.
4
Calculate the angle with tan θ and convert it to a bearing or to an angle measured from a stated direction.
5
State the final answer with both magnitude and direction, because a vector answer is incomplete without direction.
Section 4 of 5Worked Example
Question
- A runner travels 6.0 km east and then 8.0 km north.
- Find the total distance, and the displacement in size and as a bearing.
- Distance = 6.0 + 8.0 = 14.0 km because distance is a scalar and the legs simply add.
- Displacement sizeR² = 6.0² + 8.0² = 36 + 64 = 100, so R = 10 km.
- Angle from northtan θ = 6.0 ÷ 8.0 = 0.75, so θ = 36.9°, giving a bearing of 037°.
Answer
- Distance 14 km;
- displacement 10 km on a bearing of 037°.
Section 5 of 5Exam Checklist & Tips
1.1.2 – Key Points for the Exam
- Resultant The single vector that replaces several vectors and has the same effect.
- Hypotenuse The longest side of a right-angled triangle, opposite the right angle.
- Bearing A direction given as a three-figure angle clockwise from north.
- Pythagoras The relationship a² + b² = c² for right-angled triangles.
Common Mistakes
- Adding the sizes of perpendicular vectors directly instead of using Pythagoras.
- Forgetting to give a direction with the resultant, so the vector is incomplete.
- Measuring a bearing anticlockwise or from east instead of clockwise from north.
- Using the calculator in the wrong angle mode; for these calculations it must be in degrees.