# Vector analysis

• Jan 22nd 2010, 08:49 AM
yuki267
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.

• Jan 22nd 2010, 10:24 AM
arbolis
Quote:

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.

Maybe you could start by writing the components of A, for example $\displaystyle A=(a_1,a_2,a_3)$. The same for B: $\displaystyle B=(b_1,b_2,b_3)$.
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?
• Jan 22nd 2010, 10:25 AM
Plato
Quote:

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.
$\displaystyle C=\frac{A\cdot B}{A\cdot A}A~~\&~~D=B-C$
• Jan 23rd 2010, 04:46 AM
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.
• Jan 23rd 2010, 07:56 AM
arbolis
Quote:

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.