Basic Calculus help with finding components

[Zyxer22]

I always seem to find this place whenever I'm in need of homework help, so I finally decided to register (and hopefully post in the right area).

My given problem is


Here are three displacements, each in meters: d1 = 4.2i + 2.7j - 7.9k, d2 = -1.0i + 2.0j + 3.0k, and d3 = 4.0i + 3.0j + 2.0k. What is r = d1 - d2 + d3 ((a), (b) and (c) for i, j and k components respectively)? (d) What is the angle between r and the positive z axis? (e) What is the component of d1 along the direction of d2? (f) What is the component of d1 that is perpendicular to the direction of d2 and in the plane of d1 and d2?

The solutions I've gotten are

a) 9.2 m
b) 3.7 m
c) -8.9 m
d) 131.91°
e) -6.01 m
f) 7.16 m

All of which I know to be correct. The confusion I'm having is with part f. I used this equation to solve for f:

[(magnitude of d1)^2 - (the component of d1 along the direction of d2)^2]^1/2

I just don't understand why this works. Anyone help?

 

[Tomer]

This follows the pythagorian theorem. If you draw it to yourself, you'll see the the projection of d1 on d2, and d1 itself, form to vectors, which by connecting their ends form a right triangle. This is generally true for all vectors.

 

[Zyxer22]

I appreciate the answer. And, actually, now that I see it, this should've been really obvious. Thank you ^^