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

• ~angel~
In summary, the conversation covers three questions: finding a unit vector perpendicular to two given vectors, evaluating the cross product and dot product of three given vectors, and finding a vector parallel to the line of intersection of two given planes. The first question involves finding a cross product and normalizing it to get the unit vector. The second question requires performing the cross product first before the dot product. The third question involves finding a vector that is perpendicular to both given normals. The cross product of two vectors is by definition perpendicular to both of them.
~angel~

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.

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

Last edited:
Thanks.

~angel~ said:
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?

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.

OlderDan said:
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.

~angel~ said:
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.

Last edited:
So a X (b + c)?

~angel~ said:
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.

Yep. Thanks for that.

## 1. What is a unit vector?

A unit vector is a vector with a magnitude of 1. It is used to indicate direction without any specific length.

## 2. How do you find a unit vector?

To find a unit vector, divide the original vector by its magnitude. This will result in a new vector with a magnitude of 1.

## 3. What is the importance of unit vectors in physics?

Unit vectors are important in physics because they allow us to simplify calculations and represent direction without worrying about magnitude.

## 4. What is a vector v?

A vector v is a mathematical representation of a quantity that has both magnitude and direction. It is typically denoted by an arrow and can be represented by multiple components.

## 5. How do you use vectors and planes to solve problems?

Vectors and planes can be used to solve problems by representing physical quantities, such as forces and velocities, and using mathematical operations to manipulate and analyze them. They can also be used to visualize and understand geometric relationships between objects in space.

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