Calculate $\operatorname{proj}_{v} v_{0}$ Where $v=\langle 4,-6\rangle$ and $w=\langle 4,3\rangle$.
Select one:
a. $\left\langle-\frac{20}{7},-\frac{75}{26}\right\}$
b. $\left\langle\frac{10}{13},-\frac{6}{25}\right\rangle$
c. $\left\langle-\frac{8}{25}, \frac{15}{26}\right\rangle$
d. $\left\langle\frac{10}{13}, \frac{15}{26}\right\rangle$
e. $\left\langle-\frac{8}{25},-\frac{6}{25}\right\rangle$

Step 1 ：We have that

Step 2 ：\[\operatorname{proj}_{v} w = \frac{v \cdot w}{v \cdot v} v\]

Step 3 ：Substitute the given vectors into the formula

Step 4 ：\[\operatorname{proj}_{v} w = \frac{\langle 4,-6 \rangle \cdot \langle 4,3 \rangle}{\langle 4,-6 \rangle \cdot \langle 4,-6 \rangle} \langle 4,-6 \rangle\]

Step 5 ：Calculate the dot product of the vectors

Step 6 ：\[\operatorname{proj}_{v} w = \frac{16 - 18}{16 + 36} \langle 4,-6 \rangle\]

Step 7 ：Simplify the fraction

Step 8 ：\[\operatorname{proj}_{v} w = \frac{-2}{52} \langle 4,-6 \rangle\]

Step 9 ：Calculate the scalar multiplication

Step 10 ：\[\operatorname{proj}_{v} w = \langle -\frac{8}{52}, \frac{12}{52} \rangle\]

Step 11 ：Simplify the vector

Step 12 ：\[\operatorname{proj}_{v} w = \boxed{\langle -\frac{2}{13}, \frac{3}{13} \rangle}\]