Question

What are the magnitude and direction of w = ❬–5, –14❭? Round your answer to the thousandths place.

Answer 1

To solve the question, we will have to determine the quadrant within which the point falls

Since both values are negative

`(x,y)=(-5,-14)`

So the values are in the third quadrant

So we will have to get the magnitude first

`\begin{gathered} \text{Magnitude}=\sqrt[]{x^2+y^2} \\ \text{Magnitude}=\sqrt[]{(-5)^2+(-14)^2} \\ \text{Magnitude}=\sqrt[]{25+196} \\ \text{Magnitude}=221 \\ \text{Magnitude}=14.866 \end{gathered}`

Next, we will have to get the direction

`\begin{gathered} \text{direction}=\tan ^(-1)((y)/(x)) \\ \text{direction}=\tan ^(-1)((-14)/(-5)) \\ \text{direction}=\tan ^(-1)(2.8) \\ \text{direction}=70.346^0 \end{gathered}`

So, since we know that the point is on the third quadrant, then

We will add 180 degrees (90 degrees from first and 90 degrees from the second quadrant)

So we will have

`180^0+70.346^0=250.346^0`

Thus the answer is

`\mleft\Vert w\mleft\Vert=\mright?\mright?14.866,\theta=250.346^0`