I have a math assignment due tomorrow and I want to understa

I have a math assignment due tomorrow and I want to understand these two questions:

First off, three vectors A, B, and C are given by

A = 6i + 4j + 3k   B = 2i - 3j - 3k C = i + 3j - 3k

1. Decide if the vectors are parallel, orthogonal, or neither using the cross product.

2. Find a vector orthogonal to both A and the vector projection of B onto A. What is the length of this vector?

I got [-54/61, -36/61, and -27/61] for the vector projection of B onto A by the way (-9/61) * [6, 4, 3]

Thanks and sorry if it\'s in the wrong section!

Solution

1)

given A = 6i + 4j + 3k , B = 2i - 3j - 3k , C = i + 3j - 3k

A.B=-9

AxB=<-3,24,-26>

A.C=9

AxC=<-21,21,14>

B.C=2

BxC=<18,3,9>

neithor of the vectors are mutually parallel or perpendicular

2)

vector projection of B onto A = (B.A)*A_/|A|²

vector projection of B onto A = (-9)*(6i+4j+3k)_{/(6²+4²+3²)}

vector projection of B onto A = (-9_/61)*(6i+4j+3k)

vector projection of B onto A = ((-54_/61)i+(-36_/61)j+(-27_/61)k)

vector orthogonal to both A and the vector projection of B onto A = ((-54_/61)i+(-36_/61)j+(-27_/61)k)x( 6i + 4j + 3k )

i....................j..................k

(-54_/61)...... (-36_/61).......(-27_/61)

6.................4.....................3

vector orthogonal to both A and the vector projection of B onto A =(((-36_/61)*3)-(4*(-27_/61)))i+((6*(-27_/61))-((-54_/61)*3))j+(((-54_/61)*4)-(6*(-36_/61)))k

vector orthogonal to both A and the vector projection of B onto A =0i+0j+0k

length of this vector=0