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 :

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:

Task!

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

1. $3\underset{̲}{a}+7\underset{̲}{a}$
2. $2\left(7\underset{̲}{b}\right)$
3. $4\underset{̲}{q}+4\underset{̲}{r}$

Answer

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 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 $\stackrel{̂}{\underset{̲}{a}}$ .

Exercises

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

Answer