Question

Given w = −168i −160j, what are the magnitude and direction of −4w? Round the answers to the nearest whole number.

Answer 1

The magnitude and direction of −4w are 928 and 224°, respectively.

To find the magnitude and direction of −4w, we can first find the magnitude and direction of w. The magnitude of w is given by the Pythagorean theorem:

`|w| = √(((-168)^2 + (-160)^2)) = 232`

The direction of w is given by the following equation:

`tan \theta = (-160)/(-168)` = 0.947

Solving for θ, we get:

`\theta = 44^\circ`

Therefore, the magnitude and direction of w are 232 and 44°, respectively.

To find the magnitude and direction of −4w, we simply multiply the magnitude of w by −4 and add 180° to the direction of w.

The magnitude of -4w is
`232 * -4` = -928.

The direction of -4w is
`44^\circ + 180^\circ = 224^\circ`.

Answer 2

Given:

There are given the vector to find the magnitude:

`w=-168i-160j`

Step-by-step explanation:

To find the magnitude, first, we need to find the vector for -4w.

Then,

From the given vector:

`\begin{gathered} w=-168\imaginaryI-160j \\ -4w=-4(-168\mathrm{i}-160j) \\ -4w=672i+640j \end{gathered}`

Then,

The magnitude of the given vector is:

`\begin{gathered} |-4w|=√((672)^2+(640)^2) \\ =√(451584+409600) \\ =√(861184) \\ =928 \end{gathered}`

Now,

For the direction of the vector:

`\theta=tan^(-1)((y)/(x))`

Then,

`\begin{gathered} \theta=tan^(-1)((y)/(x)) \\ \theta=tan^(-1)((640)/(672)) \end{gathered}`

Then,

`\begin{gathered} \theta=tan^(-1)((640)/(672)) \\ \theta=44^(\circ) \end{gathered}`

Final answer:

The magnitude and direction of the given vector is shown below:

`\begin{gathered} magnitude:928 \\ direction:44^(\circ) \end{gathered}`

Hence, the correct option is D.