Multiplication of a Vector by a Scalar
Big Idea:
When we multiply a vector by a scalar (a regular number), we stretch, shrink, or reverse the vector — but we never change its direction (unless we flip it).
📐 Formal Definition
Let $\vec{a}$ be a vector and $k$ be a scalar (a real number). The scalar multiple $k\vec{a}$ is a vector such that:
- Magnitude: $|k\vec{a}| = |k| \cdot |\vec{a}|$
- Direction:
- Same as $\vec{a}$ if $k > 0$
- Opposite to $\vec{a}$ if $k < 0$
- Zero vector $\vec{0}$ if $k = 0$
In component form, if $\vec{a} = (a_x,\, a_y)$, then:
$$k\vec{a} = (k \cdot a_x,\, k \cdot a_y)$$
📏 Key Properties
| Property | Formula |
|---|---|
| Associativity | $m(n\vec{a}) = (mn)\vec{a}$ |
| Distributive (scalar) | $(m + n)\vec{a} = m\vec{a} + n\vec{a}$ |
| Distributive (vector) | $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$ |
| Identity | $1 \cdot \vec{a} = \vec{a}$ |
| Zero scalar | $0 \cdot \vec{a} = \vec{0}$ |
| Negation | $(-1)\vec{a} = -\vec{a}$ |