Calculus 3 - Vector Projections

[calcphys92]

Homework Statement

In three dimensions, consider the vector V = a₁i + a₂j +a₃k. Determine the projections of V onto the x, y, z axis.

Homework Equations

These are formulas from my textbook related to projection:

All underscores mean subscript.

Proj_A B = (B * A/|A|) A/|A| = ((B * A)/(A * A)) A

B*A = a_1b_1 + a_2b_2 + a_3b_3

Note: The asterisk * in the equation above is the 'dot' used in vector dot products.

PS. Sorry for not using the latex coding to make the equations look nicer. I've used this before and I know how to use the codes but when I submit them the images are broken.

The Attempt at a Solution

I don't think I'm even close but here's what I did:

(B*A / A*A) A = (a_1 / 1) j = a_1i

That's for the x axis. The projection answers for the other axes I get a_2j and a_3k respectively.

 

[tiny-tim]

welcome to pf!

hi calcphys92! welcome to pf!

Proj_A B = (B * A/|A|) A/|A| = ((B * A)/(A * A)) A
…
(B*A / A*A) A = (a_1 / 1) j = a_1i

That's for the x axis. The projection answers for the other axes I get a_2j and a_3k respectively.

yes

but that definition is a bit complicated, and difficult to remember

it's much easier to say that to find the projection on A, use e_A, the unit vector in the A direction …

then Proj_AB = (B.e_A)e_A

 

[calcphys92]

Thanks for the confirmation and advice. Also can you explain to me what the answer actually means? I'm asked "How do you interpret the results?" But I don't exactly know what vector projections actually are. Thanks in advance

 

[tiny-tim]

I'm asked "How do you interpret the results?"

well, i suppose the projection is the amount of it in that direction

or the three projections are the components that make up the original vector