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Scalar Multiplication

The Definition of Scalar Multiplication

Let $k$ be a real number and $\vec{a} = (a_1, a_2, a_3)$ . Then:

k\vec{a} = (ka_1, ka_2, ka_3)

Scalar multiplication stretches or compresses the vector $\vec{a}$ by a factor of $k$ .

Geometric interpretation

When $k > 1$ , the vector is stretched
When $0 < k < 1$ , the vector is shortened
When $k = -1$ , the vector points in the opposite direction
When $k = 0$ , you get the zero vector

Operational Properties

Distributivity: $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$

Distributivity (over scalar addition): $(k + l)\vec{a} = k\vec{a} + l\vec{a}$

Associativity of multiplication: $k(l\vec{a}) = (kl)\vec{a}$

The identity element of scalar multiplication: $1 \cdot \vec{a} = \vec{a}$

Special Cases

$(-1)\vec{a} = -\vec{a}$ (the opposite vector)
$0 \cdot \vec{a} = \vec{0}$ (the zero vector)
$\frac{1}{|\vec{a}|} \vec{a}$ (the unit vector)

The Geometric Meaning

The Stretching of a Vector

Scalar multiplication changes the length of a vector but not its direction (unless $k < 0$ ):

$k > 0$ : the direction is kept and the length becomes $k$ times the original
$k < 0$ : the direction is reversed and the length becomes $|k|$ times the original

The Change of Direction

multiplication by a positive number keeps the original direction
multiplication by a negative number reverses the direction

Worked Examples

Velocity Vectors

If an object moves at speed $v$ in some direction, then after $t$ seconds its displacement is:

$\vec{s} = v t \cdot \vec{e}$

where $\vec{e}$ is the unit vector in the direction of the velocity.

Amplifying a Force

A force $\vec{F}$ amplified by a factor of $k$ becomes $k\vec{F}$ .

Exercises

Exercise 1

Given $\vec{a} = (3, 4, 5)$ , find $2\vec{a}$ , $-\vec{a}$ and $\frac{1}{2}\vec{a}$ .

Reference Answer

scalar multiplicationscaling a vectorcoordinate computation

Approach:

Use the coordinate rule for scalar multiplication.

Steps:

$2\vec{a} = (6, 8, 10)$
$-\vec{a} = (-3, -4, -5)$
$\frac{1}{2}\vec{a} = (\frac{3}{2}, 2, \frac{5}{2})$

Answer: $2\vec{a} = (6, 8, 10)$ , $-\vec{a} = (-3, -4, -5)$ , and $\frac{1}{2}\vec{a} = (\frac{3}{2}, 2, \frac{5}{2})$

Exercise 2

Given the vector $\vec{a} = (1, 2, 3)$ , find the vector in the same direction as $\vec{a}$ with magnitude 5.

Reference Answer

scalar multiplicationthe unit vectorscaling a vector

Approach:

First find the unit vector, and then multiply it by the required length.

Steps:

$|\vec{a}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}$
The unit vector $\vec{e} = \frac{\vec{a}}{|\vec{a}|} = (\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}})$
The required vector $\vec{b} = 5\vec{e} = (\frac{5}{\sqrt{14}}, \frac{10}{\sqrt{14}}, \frac{15}{\sqrt{14}}) = \frac{5}{\sqrt{14}}(1, 2, 3)$

Answer: the required vector is $\frac{5}{\sqrt{14}}(1, 2, 3)$ , that is, $(\frac{5}{\sqrt{14}}, \frac{10}{\sqrt{14}}, \frac{15}{\sqrt{14}})$

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$k\vec{a}$	Scalar multiplication	k times a	$k$ times the vector $\mathbf{a}$
$-\vec{a}$	Negative vector	negative a	The vector in the direction opposite to $\mathbf{a}$

Vector Subtraction

Linear Combinations

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