Find a unit vector

[ibysaiyan]

1. The problem statement, all variables and given/known data
Two vectors are given by the relations

a = 2i-3j-3k
b = 6i+2j+k

Find unit vectors corresponding to a 3- dimensional orthogonal right handed coordinate system where one of the axes is parallel to _{}a and another of the axes is perpendicular to _{}b

2. Relevant equations


Cross rule: AXB = C
3. The attempt at a solution


I know from cross product that if we have two vectors in a plane then their multiple vector will also be perpendicular to them. So my vector AXB is : 3i-20j+22k ( perpendicular to a and b )

Now I know they want unit vector which is : v/ magnitude of v but in the above case I was told to find the vector : Ax(AXB)...
I am confused about this point. Can anyone clarify for me. Thanks

[HallsofIvy]

You are given a and b and asked to find "one unit vector that is parallel to a"- that should be easy- and another that perpendicular to b. As you say, axb is perpendicular to both a and b and so to b. You are then asked to find a third vector, c, say, that is perpendicular to both a and the new vector you just calculated. Yes, that is ax(axb). There is no problem with that- just do it! First find v= axb, then use that to find c= axv. I recommend first finding the vectors v and c and then dividing a, v, and c by their lengths to get unit vectors.