Multiplying a vector by a scalar
λa⃗ has magnitude |λ| |a⃗| and points along a⃗ if λ > 0, opposite if λ < 0. So b⃗ = λa⃗ means a⃗ and b⃗ are collinear.
What λa⃗ means
Multiplying a vector a⃗ by a real number λ (a scalar) gives a new vector λa⃗. Its length is |λ| times the length of a⃗. If λ is positive, λa⃗ points the same way as a⃗; if λ is negative, it points the opposite way. So 3a⃗ is three times as long as a⃗, and −2a⃗ is twice as long but reversed. In components, λ(a₁î + a₂ĵ + a₃k̂) = λa₁î + λa₂ĵ + λa₃k̂.
Properties
- k(a⃗ + b⃗) = ka⃗ + kb⃗
- (k + m)a⃗ = ka⃗ + ma⃗
- k(ma⃗) = (km)a⃗
- 1·a⃗ = a⃗ and (−1)a⃗ = −a⃗
Unit vector and a vector of given length
- 1Find the magnitude |a⃗|.
- 2Divide: â = a⃗/|a⃗|. This has length 1 and the direction of a⃗.
- 3A vector of length k along a⃗ is kâ.
- 4A vector of length k opposite to a⃗ is −kâ.
Example 1: a vector of given magnitude
Find a vector of magnitude 14 in the direction of a⃗ = 2î − 3ĵ + 6k̂.
- |a⃗| = √(4 + 9 + 36) = √49 = 7
- â = (2î − 3ĵ + 6k̂)/7
- Required vector = 14â = 2(2î − 3ĵ + 6k̂)
- = 4î − 6ĵ + 12k̂
4î − 6ĵ + 12k̂
Example 2: collinear vectors
a⃗ = 2î + 3ĵ − k̂ and b⃗ = −4î + pĵ + qk̂ are collinear. Find p and q.
- Collinear means b⃗ = λa⃗ for some λ.
- Compare the î parts: −4 = 2λ, so λ = −2.
- ĵ parts: p = 3λ = −6.
- k̂ parts: q = −1 × λ = 2.
p = −6, q = 2 (and b⃗ = −2a⃗ points opposite to a⃗).
A map scale such as 1 cm to 1 km multiplies every real displacement by the same positive scalar. Directions stay true while lengths shrink, so the map still guides you correctly.
1 marks: “If |a⃗| = 4, find |−3a⃗|.” |−3a⃗| = |−3| × |a⃗| = 3 × 4 = 12.
2 marks: “Find a vector of magnitude 6 units opposite in direction to a⃗ = 2î − ĵ + 2k̂.” |a⃗| = 3, so â = (2î − ĵ + 2k̂)/3. Required vector = −6â = −2(2î − ĵ + 2k̂) = −4î + 2ĵ − 4k̂.
2 marks: “Find the values of λ for which λ(2î − ĵ + 2k̂) is a unit vector.” |λ(2î − ĵ + 2k̂)| = |λ| × √(4 + 1 + 4) = 3|λ| = 1, so |λ| = 1/3. Hence λ = ±1/3.
λ stretches; a minus sign flips.