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