Question

Vector u has an initial point at (−5, 2) and a terminal point at (−7, 9). Which of the following represents u in trigonometric form?

Answer 1

ANSWER:

2nd option

$u=7.28\cdot(\cos 105.945\degree i+\sin 105.945\degree j)$

STEP-BY-STEP EXPLANATION:

We have that the trigonometric form is as follows:

$u=|u|\cdot(\cos \theta i+\sin \theta j)$

The first thing is to calculate the normal of the vector u, which would be the distance between both points, like this:

$\begin{gathered} |u|=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)_{}}^2 \\ \text{ we replacing} \\ |u|=\sqrt[]{(-7-(-5))^2+(9-2)^2_{}} \\ |u|=\sqrt[]{(-7+5)^2+(7_{})^2_{}}^{}=\sqrt[]{2^2+7^2}=\sqrt[]{4+49}=\sqrt[]{53} \\ |u|=7.28 \end{gathered}$

Now, the angle is calculated as follows:

$\begin{gathered} \tan \theta=\frac{y}{x} \\ \text{ in this case:} \\ \tan \theta=\frac{y_2-y_1}{x_2-x_1}=\frac{9-2}{-7-(-5)}=\frac{7}{-7+5}=-\frac{7}{2} \\ \tan \theta=-\frac{7}{2} \\ \theta=\arctan \mleft(-\frac{7}{2}\mright) \\ \theta=-74.055\degree \\ \theta=-74.055\degree+180\degree \\ \theta=105.945\degree \end{gathered}$

Therefore, the vector u in its trigonometric form would be:

$u=7.28\cdot(\cos 105.945\degree i+\sin 105.945\degree j)$