# Thread: vector & scalar projections

1. ## 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

2. 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 "$\displaystyle \theta$". 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 $\displaystyle theta$. Since "$\displaystyle cos(\theta)$" is defined as "length of near leg over length of hypotenuse", $\displaystyle cos(\theta)$ is the magnitude of the projection of a on b divided by the magnitude of a: $\displaystyle cos(\theta)= \frac{||proj_b(a)||}{||a||}$ where "$\displaystyle proj_b(a)$" is the projection of a on b.

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

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

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

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

Of course, $\displaystyle cos(\theta)= 0$ if and only if $\displaystyle \theta= 90$ degrees (for $\displaystyle \theta$ 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).