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3-2 (b) Operations on vectors

3-2 (b) Operations on vectors

Point 5 of 8 — The 3D Coordinate System and Vectors in SpaceQuestion bank: 0 questions (0 with pictures)

05 3-2 (b) Operations on vectors

Addition

  • For A⃗=(Ax,Ay,Az) and B⃗=(Bx,By,Bz): A⃗+B⃗=(Ax+Bx, Ay+By, Az+Bz)
  • Properties: closure, commutative (A⃗+B⃗=B⃗+A⃗), associative, the zero vector 0⃗=(0,0,0) is the identity, and −A⃗=(−Ax,−Ay,−Az) is the additive inverse.

Multiplication by a real number

  • kA⃗=(kAx,kAy,kAz).
  • If k>0 the direction is unchanged; if k<0 it is reversed; the norm is multiplied by ∣k∣.
  • Properties: (k+l)A⃗=kA⃗+lA⃗, k(A⃗+B⃗)=kA⃗+kB⃗, k(lA⃗)=(kl)A⃗.
✍️ Worked examples

Example 8: For A⃗=(−1,5,2) and B⃗=(4,−1,3) find 2A⃗−3B⃗ and the vector C⃗ with 2C⃗+3A⃗=2B⃗.

Solution:

  • 2A⃗−3B⃗=(−2,10,4)−(12,−3,9)=(−14,13,−5).
  • 2C⃗=2B⃗−3A⃗=(8,−2,6)−(−3,15,6)=(11,−17,0).
  • C⃗=(112,−172,0).

Equality of vectors

  • A⃗=B⃗ if and only if Ax=Bx, Ay=By and Az=Bz.
✍️ Worked examples

Example 9: Find x, y, k if (2x+1, 5, k+4)=(1, y2−4, x+1).

Solution:

  • 2x+1=1 gives x=0.
  • y2−4=5 gives y=±3.
  • k+4=x+1=1 gives k=−3.

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