Dabour Hub EGEgyptian Curriculum 2026/2027
3-2 (c) Unit vectors and the basis i^, j^, k^

3-2 (c) Unit vectors and the basis i^, j^, k^

Point 6 of 8 — The 3D Coordinate System and Vectors in SpaceQuestion bank: 0 questions (0 with pictures)

06 3-2 (c) Unit vectors and the basis i^, j^, k^

The fundamental unit vectors

  • i^=(1,0,0), j^=(0,1,0) and k^=(0,0,1) have norm 1 along the positive x, y and z axes.
  • Every vector can be written A⃗=Axi^+Ayj^+Azk^
  • For example (−1,0,0)=−i^ and (0,0,−1)=−k^.
✍️ Worked examples

Example 11: For A⃗=2i^−3j^+k^ and B⃗=−i^−2j^ find 2A⃗−3B⃗, then compare ∥A⃗+B⃗∥ with ∥A⃗∥+∥B⃗∥.

Solution:

  • 2A⃗=4i^−6j^+2k^ and 3B⃗=−3i^−6j^, so 2A⃗−3B⃗=7i^+2k^.
  • A⃗+B⃗=i^−5j^+k^, with norm 27=33≈5.20.
  • ∥A⃗∥+∥B⃗∥=14+5≈3.74+2.24=5.98.
  • So ∥A⃗+B⃗∥<∥A⃗∥+∥B⃗∥. In general ∥A⃗+B⃗∥≤∥A⃗∥+∥B⃗∥, with equality only when the vectors point the same way.

Unit vector

  • A unit vector has norm 1.
  • The unit vector in the direction of A⃗ is u^A=A⃗∥A⃗∥
  • For A⃗=(−2,2,1): ∥A⃗∥=3, so u^A=(−23,23,13).
  • A vector of norm r in the direction of A⃗ is r u^A; the opposite direction uses −u^A.

📝 Question bank for this point

0 questions linked to this point only, 0 of them with pictures.