Why Can't Vector Projections Between Two Vectors Be the Same

Question

The question is as follows:

Give two reasons why the projection of u onto v is not the same as the projection of v onto u.

I was thinking that the directions of vectors u and v are not the same so that's one way that the projections might differ. Furthermore, the length of the vectors may not be the same as well (given the exception that u and v are unit vectors in which case their length would be 1).

Are those reasons valid? If not, how can I refine them?

Answer 1

Observe that the projection of $\vec u$ onto $\vec v$ is given by

$$\vec u \cdot\frac{\vec v}{|\vec v|}=|\vec u|\cos\theta$$

while the projection of $\vec v$ onto $\vec u$ is given by

$$\vec v \cdot\frac{\vec u}{|\vec u|}=|\vec v|\cos\theta$$

and thus in general

$$\vec u \cdot\frac{\vec v}{|\vec v|}\neq\vec v \cdot\frac{\vec u}{|\vec u|}$$

Note that equality holds when

$$|\vec u|\cos\theta=|\vec v|\cos\theta\iff (|\vec u|-|\vec v|)\cos\theta=0 \iff |\vec u|=|\vec v|\quad \lor \quad \cos\theta=0 $$

that is if and only if the vectors have the same modulus or they are orthogonal.

Answer 2

The projectoin can be the same, if they are equal. When they are not equal, you have two cases:

• they are colinear: then the two projections are the identity on both, and so different.

• they are not colinear: the projection onto $u$ is colinear with $u$, while the projection onto $v$ is colinear with $v$.