# vector & scalar projections

• Mar 6th 2010, 06:47 PM
aimango
vector & scalar projections
under what circumstances is the magnitude of proj(a onto b) equal to the magnitude of proj(b onto a)?

one of the answers is: magnitude of vector a equals magnitude of vector b, and the angle between needs to be 45 or 135. can someone tell me why the angle has to be 45 or 135?

thanks
• Mar 7th 2010, 05:37 AM
HallsofIvy
Quote:

Originally Posted by aimango
under what circumstances is the magnitude of proj(a onto b) equal to the magnitude of proj(b onto a)?

one of the answers is: magnitude of vector a equals magnitude of vector b, and the angle between needs to be 45 or 135. can someone tell me why the angle has to be 45 or 135?

thanks

To "project" vector a onto vector b, draw the two vectors "tail to tail". Call the angle between them " $\theta$". Draw a line from the tip of a perpendicular to b, extending b is necessary. The vectro from the point where a and b meet to the foot of that perpendicular is the "projection of a onto b".

You now have a right triangle where vector a is the hypotenuse and the new, projection, vector is the leg of the triangle next to angle $theta$. Since " $cos(\theta)$" is defined as "length of near leg over length of hypotenuse", $cos(\theta)$ is the magnitude of the projection of a on b divided by the magnitude of a: $cos(\theta)= \frac{||proj_b(a)||}{||a||}$ where " $proj_b(a)$" is the projection of a on b.

Multiplying both sides of that equation by ||a||, $||proj_b(a)||= ||a||cos(\theta)$.

Swapping a and b, $||proj_a(b)||= ||b||cos(\theta)$.

Those will be equal when $||a||cos(\theta)= ||b||cos(\theta)$
As long as $cos(\theta)\ne 0$ we can divide both sides by it and get $||a||= ||b||$.

That is true for any angle for which $cos(\theta)\ne 0$. The angles do NOT have to be 45 degrees or 135 degrees. Who told you they did?

Of course, $cos(\theta)= 0$ if and only if $\theta= 90$ degrees (for $\theta$ between 0 and 180 degrees). In that case, a and b also have equal projection on each other- the magnitude of each projection is 0.

That is, the projection of a on b and the projection of b on a have the same magnitude if the magnitudes of a and b are the same (and the angle doesn't matter) or if the vectors are a right angles (and the magnitudes don't matter).