Scalar Multiplication

You can add vectors together. But you can also do something simpler: take a single vector and scale it — make it longer, shorter, or flip it around. That operation is called scalar multiplication.

What is a scalar?

A scalar is just a plain number — no direction, no components, just a magnitude. The number 3 is a scalar. So is −0.5, or π. When you multiply a vector by a scalar, you're resizing the arrow.

What it does to the arrow

Positive scalar — stretch or shrink

Multiplying by 2 doubles the length. Multiplying by 0.5 halves it. The direction stays exactly the same — the arrow just gets longer or shorter.

Negative scalar — flip and scale

Multiplying by −1 reverses the direction. The arrow points the opposite way, but has the same length. Multiplying by −2 doubles the length and flips the direction.

Zero — the arrow disappears

Multiplying by 0 gives the zero vector $[0,\ 0]$. No length, no direction — just a point sitting at the origin. It's the additive identity: add it to anything and nothing changes.

The algebra

The rule is simple: multiply each component by the scalar.

So $3 \cdot [2,\ 1] = [6,\ 3]$. The vector triples in length and keeps pointing the same way.

Length scales by the absolute value

If the original vector has length $\|\mathbf{v}\|$, then the scaled vector has length $|s| \cdot \|\mathbf{v}\|$. The absolute value of the scalar tells you how much the length changes; the sign tells you whether the direction flips.

In code

import numpy as np

v = np.array([2, 2])

print(3 * v)     # [ 6  6]  — stretch
print(-1 * v)    # [-2 -2]  — flip
print(0.5 * v)   # [1. 1.]  — shrink

Why this matters

Scalar multiplication is one half of a bigger idea coming up shortly: linear combinations. When you can both scale vectors and add them together, you can reach any point in space — or describe complex motions as weighted sums of simpler ones.