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EXAMPLE 6 Find the scalar projection and vector projection of b = (4, 2, 1) onto a = (-2, 4, 3). SOLUTION Since $|a| = \sqrt{(-2)^2 + 4^2 + 3^2} = \sqrt{29}$, the scalar projection of b onto a is $comp_a b = \frac{a \cdot b}{|a|} = \frac{(-2)(4) + 4(2) + 3(1)}{\sqrt{29}}$ $= \frac{3\sqrt{29}}{29}$ The vector projection is this scalar projection times the unit vector in the direction of a, below. $proj_a b = \frac{3\sqrt{29}}{29} \cdot \frac{a}{|a|} = \frac{3\sqrt{29}}{29} \cdot a$

          EXAMPLE 6 Find the scalar projection and vector projection of b = (4, 2, 1) onto a = (-2, 4, 3).
SOLUTION Since $|a| = \sqrt{(-2)^2 + 4^2 + 3^2} = \sqrt{29}$, the scalar projection of b onto a is
$comp_a b = \frac{a \cdot b}{|a|} = \frac{(-2)(4) + 4(2) + 3(1)}{\sqrt{29}}$
$= \frac{3\sqrt{29}}{29}$
The vector projection is this scalar projection times the unit vector in the direction of a, below.
$proj_a b = \frac{3\sqrt{29}}{29} \cdot \frac{a}{|a|} = \frac{3\sqrt{29}}{29} \cdot a$

EXAMPLE 6 Find the scalar projection and vector projection of b = (4, 2, 1) onto a = (-2, 4, 3).
SOLUTION Since |a| = √((-2)^2 + 4^2 + 3^2) = √(29), the scalar projection of b onto a is
compa b = (a · b)/(|a|) = ((-2)(4) + 4(2) + 3(1))/(√(29))
= (3√(29))/(29)
The vector projection is this scalar projection times the unit vector in the direction of a, below.
proja b = (3√(29))/(29)·(a)/(|a|) = (3√(29))/(29)· a

Added by Daniel V.

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