Multiplying Vector by Scalar

A vector being scaled by a positive and a negative number

Multiplying a vector by a number changes the length, but not the direction, of the vector, unless you multiply by a negative number. In that case the vector will be rotated $180$ degrees. If you have a vector written as a letter with an arrow, it follows the regular rules of algebra, like this:

Rule

Multiplication of Numbers and Vectors

If you have a vector on vector form $\vec{v}$ , you contract the number with the vector, just like you did when you multiplied a number with a letter in algebra:

k \cdot \vec{v} = k \vec{v} .

If you have the vector on coordinate form, $\vec{v} = (x, y)$ , you can just multiply the number by all the coordinates in the vector:

k \vec{v} = k \cdot (x, y) = (k x, k y) .

Example 1

Find $4 \cdot (2, 4)$ .

4 \cdot (2, 4) = (8, 16)

Example 2

Find $- 2 \cdot (2 t, t^{2})$ .

- 2 \cdot (2 t, t^{2}) = (- 4 t, - 2 t^{2})

Example 3

$2 \cdot \vec{a}$ is a vector with the same direction as $\vec{a}$ , but it is twice as long. You write it as $2 \vec{a}$ .
$- 3 \cdot \vec{a}$ is a vector in the opposite direction of $\vec{a}$ , and it’s also three times as long as $\vec{a}$ . It is written as $- 3 \vec{a}$ .

Example 4

Find $2 \cdot (2 x, 4 y) + 3 \cdot (3 x, - y)$ .

$\begin{array}{l} 2 \cdot (2 x, 4 y) + 3 \cdot (3 x, - y) \\ = (4 x, 8 y) + (9 x, - 3 y) \\ = (13 x, 5 y) \end{array}$

$\begin{array}{l} 2 \cdot (2 x, 4 y) + 3 \cdot (3 x, - y) & = (4 x, 8 y) + (9 x, - 3 y) \\ = (13 x, 5 y) \end{array}$

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