Find a Normal Vector to Plane ax+by=cz=d | Calc III Homework

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In summary, the normal vector to any plane with the equation ax+by=cz=d is <a,b,c>. However, any scalar multiple of this vector, such as <-a,-b,-c> or <-2a,-2b,-2c>, would also be a valid normal vector.
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Homework Statement


Take any plane ax+by=cz=d (any numbers should work if subsituted)
Find a normal vector to the plane Read my question below


Homework Equations





The Attempt at a Solution


I know the normal vector would be <a,b,c>. But, could a correct answer also be a scalar multiple to <a,b,c> such as <-a,-b,-c> or <-2a,-2b,-2c>? I was thinking about this but could not come up with an answer myself!

Thanks for any help--I am working on learning Calc III myself-such an AWESOME subject!
 
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  • #2
Exactly right. Any multiple of <a,b,c> is also normal to the plane.
 
  • #3
Thanks so much for the prompt reply.
 

1. What is a normal vector?

A normal vector is a vector that is perpendicular to a given surface or plane. It represents the direction in which the surface or plane is pointing.

2. How do you find a normal vector to a plane?

To find a normal vector to a plane, you can use the coefficients of the plane's equation. For the plane ax+by=cz=d, the normal vector is given by the equation (a, b, c).

3. Can there be more than one normal vector to a plane?

Yes, there can be infinitely many normal vectors to a plane. This is because any vector that is perpendicular to the plane can be considered a normal vector.

4. How do you determine the direction of a normal vector?

The direction of a normal vector can be determined by the sign of its components. For example, if the normal vector is (a, b, c), the direction is determined by the signs of a, b, and c. If all three components are positive, the direction is in the positive direction; if all three components are negative, the direction is in the negative direction; and if the components have a mix of positive and negative values, the direction will be a combination of both.

5. Can a normal vector be used to find the angle between two planes?

Yes, the angle between two planes can be found using their respective normal vectors. The angle between two planes is equal to the angle between their normal vectors.

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