Explanation:
The scalar projection of a vector b onto a vector a is given by the dot product of b and the unit vector of a:
p = (b · u) * u
where u is the unit vector of a, and can be found by dividing a by its magnitude:
u = a / ||a||
The vector projection of b onto a is simply the scalar projection scaled by the unit vector:
p = p * u
For the given vectors a = ⟨-5,12⟩ and b = ⟨4,6⟩, we first find the unit vector u:
u = a / ||a|| = ⟨-5,12⟩ / ||⟨-5,12⟩|| = ⟨-5/13,12/13⟩
Next, we find the dot product of b and u:
p = (b · u) = ⟨4,6⟩ · ⟨-5/13,12/13⟩ = (4 * -5/13) + (6 * 12/13) = -20/13 + 72/13 = 52/13
So the scalar projection of b onto a is p = 52/13.
Finally, we find the vector projection by scaling the unit vector u by the scalar projection:
p = p * u = (52/13) * ⟨-5/13,12/13⟩ = ⟨-52/169,104/169⟩
So the vector projection of b onto a is p = ⟨-52/169,104/169⟩.