Cross/Dot Product

• Apr 3rd 2007, 09:42 AM
Raman
Cross/Dot Product
Hey, I'm stuck on one problem for my homework (LAST ONE!!) It seems really simple but I can't figure it out. Here it is:

Express the vector A = 2i - 3j + 5k as a sum A = B + C, where B is parallel to the vector V = 3i -2j +5k and C is perpendicular to V.

Here's what I know:
B x V = 0, because B is parallel to V
C (dot) V = 0, because C is perpendicular to V

However I can't find a way to apply the above :confused:

Thanks! :o
• Apr 3rd 2007, 10:39 AM
Plato
B=[(A.V)/(V.V)]V and C=A-B, where (A.V) is the dot product of A and V.
• Apr 3rd 2007, 11:31 AM
Soroban
Hello, Raman!

There are formulas for this problem . . .

Quote:

Express the vector: .A .= .2i - 3j + 5k .as a sum .A .= .B + C,
where B is parallel to the vector V .= .3i - 2j + 5k, and C is perpendicular to V.

Code:

```                          *                 A    *  :                   *      : C               *          :           *---------------+-------*           : - - - B - - - :           : - - - - - V - - - - - :```

Given vectors A and V . . . B is the projection of A onto V.

. . . . . . . . . . . . . . . . .A·V
The formula is: .B . = . ----- V
. . . . . . . . . . . . . . . . .|V

. . . . . . . . . . . . . . .[2,-3,5]·[3,-2,5] . . . . . . .37
So we have: .B . = . --------------------- V . = . --- V
. . . . . . . . . . . . . . .(√3² + 2² + 5²)² . . . . . .38

Hence: .B .= .(37/38)[3,-2,5] .= .[111/38, -74/38, 185/38]

Another formula: . C .= .A - B

Hence: .C .= .[2, -3, 5] - [111/38, -74/38, 185/38] .= .[-35/38, -40/38, 5/38]

Therefore: .A .= .[111/38, -74/38, 185/38] + [-35/38, -40/38, 5/38]

• Apr 3rd 2007, 12:33 PM
Raman
Thanks!
Wow, thanks a LOT, that picture/diagram really makes it all clear! Great explaination btw as well. Once again, thank you, both of you! :)