College Trigonometry, 2011a

11.9 The Dot Product and Projection 1041
⃗q · ⃗w = (⃗v − ⃗p) · ⃗w Definition of ⃗q.
= ⃗v · ⃗w − ⃗p · ⃗w Properties of Dot Product
( ) ⃗v · ⃗w
= ⃗v · ⃗w −
⃗w · ⃗w ⃗w · ⃗w Since ⃗p =proj ⃗w (⃗v).
( ) ⃗v · ⃗w
= ⃗v · ⃗w − ( ⃗w · ⃗w) Properties of Dot Product.
⃗w · ⃗w
= ⃗v · ⃗w − ⃗v · ⃗w
= 0
Hence, ⃗q · ⃗w = 0, as required. At this point, we have shown that the vectors ⃗p and ⃗q guaranteed
by Theorem 11.27 exist. Now we need to show that they are unique. Suppose ⃗v = ⃗p + ⃗q = ⃗p ′ + ⃗q ′
where the vectors ⃗p ′ and ⃗q ′ satisfy the same properties described in Theorem 11.27 as ⃗p and ⃗q.
Then ⃗p − ⃗p ′ = ⃗q ′ − ⃗q, so ⃗w · (⃗p − ⃗p ′ ) = ⃗w · (⃗q ′ − ⃗q) = ⃗w · ⃗q ′ − ⃗w · ⃗q = 0 − 0 = 0. Hence,
⃗w · (⃗p − ⃗p ′ ) = 0. Now there are scalars k and k ′ so that ⃗p = k⃗w and ⃗p ′ = k ′ ⃗w. This means
⃗w · (⃗p − ⃗p ′ )= ⃗w · (k⃗w − k ′ ⃗w) = ⃗w · ([k − k ′ ] ⃗w) =(k − k ′ )( ⃗w · ⃗w) =(k − k ′ )‖ ⃗w‖ 2 . Since ⃗w ≠ ⃗0,
‖ ⃗w‖ 2 ≠ 0, which means the only way ⃗w · (⃗p − ⃗p ′ )=(k − k ′ )‖ ⃗w‖ 2 =0isfork − k ′ =0,ork = k ′ .
This means ⃗p = k⃗w = k ′ ⃗w = ⃗p ′ . With ⃗q ′ − ⃗q = ⃗p − ⃗p ′ = ⃗p − ⃗p = ⃗0, it must be that ⃗q ′ = ⃗q as
well. Hence, we have shown there is only one way to write ⃗v as a sum of vectors as described in
Theorem 11.27.
We close this section with an application of the dot product. In Physics, if a constant force F is
exerted over a distance d, thework W done by the force is given by W = Fd. Here, we assume the
force is being applied in the direction of the motion. If the force applied is not in the direction of
the motion, we can use the dot product to find the work done. Consider the scenario below where
the constant force ⃗ F is applied to move an object from the point P to the point Q.
⃗F
⃗F
P
θ
Q
θ
To find the work W done in this scenario, we need to find how much of the force F ⃗ is in the
direction of the motion −→ PQ. This is precisely what the dot product F ⃗ · ̂PQ represents. Since
the distance the object travels is ‖ −→ PQ‖, wegetW =( F ⃗
−→ −→ −→
· ̂PQ)‖ PQ‖. Since PQ = ‖ PQ‖̂PQ,
W =( F ⃗ −→
· ̂PQ)‖ PQ‖ = F ⃗
−→ · (‖ PQ‖̂PQ)= F ⃗
−→ · PQ = ‖ F ⃗
−→ ‖‖ PQ‖ cos(θ), where θ is the angle between
the applied force F ⃗ and the trajectory of the motion −→ PQ. We have proved the following.