Question

Find the Projection of v onto u where u=(2/3)i-(2/3)j-(1/3)k and v=2i-2j+2k

Answer 1

The projection of a vector \(\vec{a}\) onto another vector \(\vec{b}\) is given by
\[\|a\|\cos\theta\times\frac{\vec{b}}{\|\vec{b}\|}\]
where \(\theta\) is the angle between the vectors. Recall that
\[\vec{a}\cdot\vec{b}=\|a\|\|b\|\cos\theta~~\iff~~\cos\theta=\frac{\vec{a}\cdot\vec{b}}{\|a\|\|b\|}\]

Answer 2

All this means you need to find the dot product of the given vectors along with the norms of each vector.

Answer 3

Would the opposite also be true?

Answer 4

projection of B onto A

Answer 5

By "opposite" you must mean switch \(a\) with \(b\) in the formulas above? Then yes. The projection of \(b\) onto \(a\) would be \(\|\vec{b}\|\cos\theta\dfrac{\vec{a}}{\|\vec{a}\|}\).

Answer 6

\[\theta=\cos^{-1} \frac{ UV \ }{ \left| UV \right|^2 }V\]

Answer 7

oh alright thank you

Answer 8

yw