Question

A vector $\overrightarrow{\mathbf{d}}$has a magnitude 3.0 m and is directed south. Whatare (a) the magnitude and (b) the direction of the vector$\mathbf{5}\mathbf{.}\mathbf{0}\overrightarrow{\mathbf{d}}$? Whatare (c) the magnitude and (d) the direction of the vector $\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\overrightarrow{\mathbf{d}}$?

Short answer

a) The magnitude of the vector $5.0\overrightarrow{d}$is 15 m.

b)The direction of the vector $5.0\overrightarrow{d}$is south.

c)The magnitude of the vector $-2.0\overrightarrow{d}$is 6 m.

d)The direction of the vector $-2.0\overrightarrow{d}$is north.

Step-by-step solution

To understand the concept

Using the standard concept of magnitude and direction, the magnitude and the direction of the given vector can be calculated. In this problem the vector is pointing towards south.

Thus vector $\overrightarrow{\mathrm{d}}$is represented by

$\overrightarrow{d}=\left(3m\right)\left(-\hat{j}\right)$

To find magnitude of the vector5.0d→

Using equation (i), the magnitude of $5.0\overrightarrow{d}$can be written as

$$\begin{gathered} 5.0\overrightarrow{d}=5\left(3m\right)\left(-\hat{j}\right) \\ 5.0\overrightarrow{d}\left(-15\right)\hat{j} \end{gathered}$$

It is known that positive scalars affect the direction but not the magnitude.

So the magnitude of $5.0\hat{d}$is$15\mathrm{m}$ .

To find the direction of the vector 5.0d→

Again, positive scalars affect the direction, the direction of the vector$5.0\overrightarrow{d}$ is south.

To find the direction of the vector -2.0d→

Using equation (i), the magnitude of $-2.0\overrightarrow{d}$is given by

$$\begin{gathered} -2\overrightarrow{\mathrm{d}}=\left(3\mathrm{m}\right)\left(-\hat{\mathrm{j}}\right) \\ -2\overrightarrow{\mathrm{d}}=-2\left(3\mathrm{m}\right)\left(-\hat{\mathrm{j}}\right) \\ -2\overrightarrow{\mathrm{d}}=6\mathrm{m} \end{gathered}$$

To find the magnitude of the vector -2.0d→

Here the scalar is negative. Thus the negative sign reverses the direction. Therefore, the new direction of the vector $-2.0\overrightarrow{d}$ is north.