Question

A vector $\overrightarrow{\mathbf{d}}$has a magnitude of 2.5 mand points north. What are (a) the magnitude and (b) the direction of role="math" localid="1660888359779" $\mathbf{4}\mathbf{.}\mathbf{0}\overrightarrow{\mathbf{d}}$? What are (c) the magnitude and (d) the direction ofrole="math" localid="1660888453551" $\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{0}\overrightarrow{\mathbf{d}}$?

Short answer

1. Magnitude of $4.0\overrightarrow{\mathrm{d}}$is 10.0 m
2. Direction of $4.0\overrightarrow{\mathrm{d}}$is towards north
3. Magnitude of role="math" localid="1660888804173" $-3.0\overrightarrow{\mathrm{d}}$is 7.5 m
4. Direction of $-3.0\overrightarrow{\mathrm{d}}$is towards south

Step-by-step solution

Given data

The magnitude of$\overrightarrow{d}$ is 2.5 m and direction is towards north.

Understanding the concept

Vectors can be multiplied by the scalars. If we multiply the vectors by positive scalar, the direction will be same. If we multiply vectors with negative scalar, the direction will be reversed.

Formula:

$\overrightarrow{A}=n·\overrightarrow{d}$(A would be in the same direction of d but n times greater)

$\overrightarrow{A}=\left(-n\right)·\overrightarrow{d}$(A would be n times greater than d but opposite in the direction of d)

(a) Calculate the magnitude of

Use the given value of the magnitude of vector $\overrightarrow{d}$to calculate the magnitude of role="math" localid="1660888706506" $4.\overrightarrow{d}$

$\begin{array}{rcl}\left|4·\overrightarrow{d}\right|& =& 4·\left|\overrightarrow{d}\right|\\ & =& 4\times 2.5\text{m}\\ & & \text{= 10}\text{m}\\ & & \\ & & \end{array}$

Therefore, the magnitude of $4.\overrightarrow{d}$is 10 m.

(b) Calculate the direction of 4.0d→

Since, the vector is multiplied by the positive scalar, the direction will remain the same, i.e., towards north.

(c) Calculate the magnitude of -3.0d→

Use the given value of the magnitude of vector $\overrightarrow{d}$to calculate the magnitude of $-3.0\overrightarrow{d}$

$\begin{array}{rcl}\left|-3·\overrightarrow{d}\right|& =& 3·\left|\overrightarrow{d}\right|\\ & =& 3\times 2.5\text{m}\\ & =& 7.5\text{m}\\ & & \end{array}$

Therefore, the magnitude of $-3.0\overrightarrow{d}$is 7.5 m.

(d) Calculate the direction of -3.0d→

The negative sign indicates that the direction will be opposite to the direction of $\overrightarrow{d}$. So, it will be towards south.