Thread: Vector analysis

Hi, I am having trouble with this problem :

Let A and B be 2 vectors. Express the vector B as the sum of a vector C, that is parallel to A, and of a vector D that is perpendicular to A. Note : This is in 3-D space.

I don't even know how to start.

Thanks in advance

Originally Posted by yuki267

Hi, I am having trouble with this problem :

Let A and B be 2 vectors. Express the vector B as the sum of a vector C, that is parallel to A, and of a vector D that is perpendicular to A. Note : This is in 3-D space.

I don't even know how to start.

Thanks in advance

Maybe you could start by writing the components of A, for example . The same for B: .
Do the same for C and D.
Now, C is parallel to A. It means that their cross product is null. (see Cross Product -- from Wolfram MathWorld). So do the cross product between those 2 vectors and equal it to 0.
D is orthogonal to A, it means that their dot product is null.
Can you take it from there?

Originally Posted by yuki267

Let A and B be 2 vectors. Express the vector B as the sum of a vector C, that is parallel to A, and of a vector D that is perpendicular to A. Note : This is in 3-D space.

Try these.

What they area asking you to do is to find the "parallel projection" and "orthogonal projection" of A on B. Plato gave you the formulas for that.

Originally Posted by HallsofIvy

What they area asking you to do is to find the "parallel projection" and "orthogonal projection" of A on B. Plato gave you the formulas for that.

Oh I missed this. I think my method still works but I agree it's much longer than Plato's one.