What is the projection of <3,4,−1> onto <−1,3,−6>?

1 Answer
JMicheli
Jul 12, 2016

The projection is (−15√46,45√46,−90√46).

Explanation:

Given two vectors →V and −→W, the projection of →V onto −→W is given by:

(→V⋅−→W)−→W∣∣∣∣∣∣−→W∣∣∣∣∣∣

The inner product gives the component of VecV in the direction of VecW, and the fraction performs the same function as multiplying this magnitude by a unit vector in the direction of −→W.

So plugging in given values:
→V=(3,4,−1)
−→W=(−1,3,−6)
∣∣∣∣∣∣−→W∣∣∣∣∣∣=√1+9+36=√46

→V⋅−→W=(3⋅−1)+(4⋅3)+(−1⋅−6)
=−3+12+6=15

So plugging everything in, the projection is:
15⋅(−1,3,−6)√46=(−15√46,45√46,−90√46)

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