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Scalar Multiplication of Vectors

To multiply a vector by a scalar, multiply each component by the scalar.

If $\vec{u} = ⟨ u_{1}, u_{2} ⟩$ has a magnitude $| \vec{u} |$ and direction $d$ , then $n \vec{u} = n ⟨ u_{1}, u_{2} ⟩ = ⟨ n u_{1}, n u_{2} ⟩$ where $n$ is a positive real number, the magnitude is $| n \vec{u} |$ , and its direction is $d$ .

Note that if $n$ is negative, then the direction of $n u$ is the opposite of $d$ .

Example :

Let $u = ⟨ - 1, 3 ⟩$ , Find $7 u$ .

$\begin{array}{l} 7 u = 7 ⟨ - 1, 3 ⟩ \\ = ⟨ 7 (- 1), 7 (3) ⟩ \\ = ⟨ - 7, 21 ⟩ \end{array}$

Properties of Scalar Multiplication:

Let $u$ and $v$ be vectors, let $c$ and $d$ be scalars. Then the following properties are true.

Properties of Scalar Multiplication
The magnitude of the scaled vector is equal to the absolute value of the scalar times the magnitude of the vector.	$‖ c v ‖ = \| c \| v$
Associative Property	$c (d u) = (c d) u$
Commutative Property	$c u = u c$
Distributive Property	$(c + d) u = c u + d u$ $c (u + v) = c u + c v$
Identity Property	$1 \cdot u = u$
Multiplicative Property of $- 1$	$(- 1) c = - c$
Multiplicative Property of $0$	$0 (u) = 0$