Question

How do I find the direction angle of a vector < -2, -5 > ?

Answer 1

Hint: For solving this problem we will use the formula of \[\tan \theta = \dfrac{y}{x}\] . Here the $\theta $ will give us the angle of the vector and x is horizontal change and y is vertical change. Moreover, we have to transfer tan to the other side of equals to that is to the right side you need to take.

Complete step by step solution:
The given problem statement is to find the direction angle of the vector < -2, -5 >.
For the given problem statement we will use the formula \[\tan \theta = \dfrac{y}{x}\] . Let us suppose that we are looking for the angle with the x-axis which the vector makes.
\[ \Rightarrow \tan \theta = \dfrac{y}{x}\]
Moreover, we have to transfer tan to the other side of equals to that is to the right side you need to take, we get,
$ \Rightarrow \theta = {\tan ^{ - 1}}(\dfrac{y}{x})$
After substituting the values that is y=-5 and x=-2 in the above equation which we have, hence, we will get,
 $ = {\tan ^{ - 1}}(\dfrac{{ - 5}}{{ - 2}})$
Now, both the minus in the numerator and denominator gets cancelled, and we get,
 $ = {\tan ^{ - 1}}(\dfrac{5}{2})$
After dividing 5 with 2 we get,
 $ = {\tan ^{ - 1}}(2.5)$
By calculating this we will get the value for${\tan ^{ - 1}}(2.5)$ in degrees \[{68.199^ \circ }\] or in radians is 1.19.

So, vector <-2, -5> is at an angle of \[{68.199^ \circ }\] to the negative x-axis.

Additional Information:
In the above question we have calculated both the radian and degree. Also, remember that 1 radian = $\dfrac{{180}}{\pi }$ degree. And we can easily convert radian to a degree by multiplying radian to $\dfrac{{180}}{\pi }$ . For converting degrees to radians we just have to multiply degrees with $\dfrac{\pi }{{180}}$ .

Note:
In the above question we were asked to find the direction angle and that’s why we have used the formula $\tan \theta = \dfrac{x}{y}$ . And since we only have to find the angle that is $\theta $ so we transferred tan on the other side of equals to. Also, we need to understand that this angle is in the third quadrant because we have -2 and -5 which means both x-axis and y-axis is negative and that happens only in the third quadrant.