Multiplying a vector by a scalar

5 Multiplying a vector by a scalar

If $k$ is any positive scalar and $\underset{̲}{a}$ is a vector then $k \underset{̲}{a}$ is a vector in the same direction as $\underset{̲}{a}$ but $k$ times as long. If $k$ is negative, $k \underset{̲}{a}$ is a vector in the opposite direction to $\underset{̲}{a}$ and $k$ times as long. See Figure 19. The vector $k \underset{̲}{a}$ is said to be a scalar multiple of $\underset{̲}{a}$ .

Figure 19 :

{ Multiplying a vector by a scalar}

The vector $3 \underset{̲}{a}$ is three times as long as $\underset{̲}{a}$ and has the same direction. The vector $\frac{1}{2} \underset{̲}{r}$ is in the same direction as $\underset{̲}{r}$ but is half as long. The vector $- 4 \underset{̲}{b}$ is in the opposite direction to $\underset{̲}{b}$ and four times as long.

For any scalars $k$ and $l$ , and any vectors $\underset{̲}{a}$ and $\underset{̲}{b}$ , the following rules hold:

Key Point 2

k (\underset{̲}{a} + \underset{̲}{b}) = k \underset{̲}{a} + k \underset{̲}{b}

(k + l) \underset{̲}{a} = k \underset{̲}{a} + l \underset{̲}{a}

k (l \underset{̲}{a}) = (k l) \underset{̲}{a}

Task!

Using the rules given in Key Point 2, simplify the following:

$3 \underset{̲}{a} + 7 \underset{̲}{a}$
$2 (7 \underset{̲}{b})$
$4 \underset{̲}{q} + 4 \underset{̲}{r}$

Using the second rule, $3 \underset{̲}{a} + 7 \underset{̲}{a}$ can be simplified to $(3 + 7) \underset{̲}{a} = 10 \underset{̲}{a}$ .
Using the third rule $2 (7 \underset{̲}{b}) = (2 \times 7) \underset{̲}{b} = 14 \underset{̲}{b}$ .
Using the first rule $4 \underset{̲}{q} + 4 \underset{̲}{r} = 4 (\underset{̲}{q} + \underset{̲}{r})$ .

5.1 Unit vectors

A vector which has a magnitude of 1 is called a unit vector . If $\underset{̲}{a}$ has magnitude 3, then a unit vector in the direction of $\underset{̲}{a}$ is $\frac{1}{3} \underset{̲}{a}$ , as shown in Figure 20.

Figure 20 :

{ A unit vector has length one unit}

A unit vector in the direction of a given vector is found by dividing the given vector by its magnitude:

A unit vector in the direction of $\underset{̲}{a}$ is given the ‘hat’ symbol $\hat{\underset{̲}{a}}$ .

Key Point 3

A unit vector can be found by dividing a vector by its modulus.

\hat{\underset{̲}{a}} = \frac{\underset{̲}{a}}{| \underset{̲}{a} |}

Exercises

Draw an arbitrary vector $\underset{̲}{r}$ . On your diagram draw $2 \underset{̲}{r}$ , $4 \underset{̲}{r}$ , $- \underset{̲}{r}$ , $- 3 \underset{̲}{r}$ and $\frac{1}{2} \underset{̲}{r}$ .
In triangle $O A B$ the point $P$ divides $A B$ internally in the ratio $m : n$ . If $\vec{O A} = \underset{̲}{a}$ and $\vec{O B} = \underset{̲}{b}$ depict this on a diagram and then find an expression for $\vec{O P}$ in terms of $\underset{̲}{a}$ and $\underset{̲}{b}$ .

$\vec{O P} = \underset{̲}{a} + \frac{m}{m + n} (\underset{̲}{b} - \underset{̲}{a}) = \frac{n \underset{̲}{a} + m \underset{̲}{b}}{m + n}$ .