Calculus- Early Transcendentals, 2021a

430 Three Dimensions
v
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θ .
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w
Figure 12.5: p is the projection of v onto w.
Using a little trigonometry, we see that
|p| = |v|cosθ = |v| v · w
|v||w| = v · w
|w| ;
this is sometimes called the scalar projection of v onto w. Togetp itself, we multiply this length by a
vector of length one parallel to w:
p = v · w w
|w| |w| = v · w
|w| 2 w.
Be sure that you understand why w/|w| is a vector of length one (also called a unit vector) parallel to w.
The discussion so far implicitly assumed that 0 ≤ θ ≤ π/2. If π/2 < θ ≤ π, the picture is like Figure
12.6. In this case v · w is negative, so the vector
v · w
|w| 2 w
is anti-parallel to w, and its length is
∣ ∣∣∣ v · w
|w| ∣ .
In general, the scalar projection of v onto w may be positive or negative. If it is negative, it means that
the projection vector is anti-parallel to w and that the length of the projection vector is the absolute value
of the scalar projection. Of course, you can also compute the length of the projection vector as usual, by
applying the distance formula to the vector.
v
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θ
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w
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p
Figure 12.6: p is the projection of v onto w.
Note that the phrase “projection onto w” is a bit misleading if taken literally; all that w provides is a
direction; the length of w has no impact on the final vector. In Figure 12.7, for example, w is shorter than
the projection vector, but this is perfectly acceptable.