Multiplying a vector by a number, equal vectors and unit vectors
Multiplying a vector by a real number k (a scalar) multiplies each component: k(Aₓ, Aᵧ, Az) = (kAₓ, kAᵧ, kA_z). The result kA is parallel to A. Its norm is ‖kA‖ = |k|·‖A‖. If k is positive it points the same way as A; if k is negative it points the opposite way; k = 0 gives the zero vector. The table shows A = (2, −1, 2), whose norm is 3. These rules hold: k(A + B) = kA + kB, (k + m)A = kA + mA and k(mA) = (km)A. Subtraction is addition of the opposite: A − B = A + (−1)B. Two vectors are equal when all three components are equal, so one vector equation is really three number equations, which is how we find unknown letters. A unit vector (in Arabic متجه الوحدة) has norm 1. Dividing a vector by its own norm shrinks it to length 1 without changing its direction, so A/‖A‖ is the unit vector that points the same way as A. Then a vector of norm m in that direction is m·A/‖A‖.
| k | kA for A = (2, −1, 2) | ‖kA‖ | compared with A |
|---|---|---|---|
| 2 | (4, −2, 4) | 6 | same direction, twice as long |
| −1 | (−2, 1, −2) | 3 | opposite direction, same length |
| 1/3 | (2/3, −1/3, 2/3) | 1 | same direction, length 1: a unit vector |
| −1/2 | (−1, 1/2, −1) | 3/2 | opposite direction, half as long |
Stretch and flip. Take A = (1, 2, 2) (norm 3). Write kA for k = 2, −1, 1/3 and −3 and say in each case whether the direction is kept and what the norm becomes.