Help Needed! Vectors & Planes: Find Unit Vector & Vector v

[~angel~]

Please help.

Given the 3 vectors a = -21 + 3j - k, b = 41 - j + 2k and c = -3i + 2j - 3k:

1. Find the unit vector perpendicular to a and b + c.

2. Evaluate a . b x c

I'm completely clueless on how to approach the first question. Any help would be great.

I'm not sure which product I'm meant to perform first in the second question.

Also,

3. p1 and p2 are planes with cartesian equations 2x - y + 3z = 5 and
x - 3y + z = -2, respectively, and l is the line of intersection of p1 and p2.

Find a vector v parallel to l.

I've already determined the normals of both planes:

for p1 : 2i - j + 3k and for p2 : i - 3j + k,

but I'm not sure where to go from here. Clearly v will be perpendicular to both normals, but I don't know how to find that vector.

Any help for these questions would be greatly appreciated.

 

[OlderDan]

Several questions by several posters, so let me just get you started on the first one. I assume you can add b + c. A vector perpendicular to a and b + c is the cross product of a with the sum of b + c. The unit vector is found by dividing that vector by its length.

In the second question, it only makes sense if you do the cross product first. If you did a . b there would be no vector to cross with c

 

[~angel~]

Thanks.

 

[OlderDan]

Thanks.

I think you can get 3 now. You are right that v is perpendicular to both normals, and you know the normals. So what vector do you know for sure is perpendicular to both of them?

 

[~angel~]

If you have time, could you tell me how you know that the cross product of b and c is perpendicular to both a and b + c? Thank you.

 

[~angel~]

I think you can get 3 now. You are right that v is perpendicular to both normals, and you know the normals. So what vector do you know for sure is perpendicular to both of them?

I bet his question is really easy, but I can't seem to get it.

 

[OlderDan]

If you have time, could you tell me how you know that the cross product of b and c is perpendicular to both a and b + c? Thank you.

I stated it incorrectly. I will go back and edit it. What I meant to say was

The vector that is perpendicular to a and b + c is the cross product of the vector a with the vector that is the sum of the two vectors b + c.

 

[~angel~]

So a X (b + c)?

 

[OlderDan]

So a X (b + c)?

Yes. That is it. Then you have to normalize it to get the unit vector. The cross product is by definition perpendicular to the two vectors in the product.

 

[~angel~]

Yep. Thanks for that.