# Finding vector perpendicular to another vector

• TsAmE
In summary, to find a vector perpendicular to (2,-1,1), you can use the dot product and set it equal to 0, resulting in a single equation in three unknown values. From there, you can either choose values for two of the variables and solve for the third, or express z as a function of x and y to find all vectors perpendicular to (2,-1,1).
TsAmE

## Homework Statement

How would you find a vector perpendicular to (2,-1,1)?

None.

## The Attempt at a Solution

I tried saying (x,y,z) x (2,-1,1) = 1 (since sin90 max at 1)

but have 3 unknowns, x, y and z so don't know how to get a numerical answer

TsAmE said:

## Homework Statement

How would you find a vector perpendicular to (2,-1,1)?

None.

## The Attempt at a Solution

I tried saying (x,y,z) x (2,-1,1) = 1 (since sin90 max at 1)

but have 3 unknowns, x, y and z so don't know how to get a numerical answer

When you cross two vectors, the result should also be a vector, so writing (x,y,z) x (2,-1,1) = 1 doesn't really make sense.

Consider using the dot product.

There exist an infinite number of vectors (an entire plane) perpendicular to a given vector in three dimensions. If you want to use the cross product, then it would be the magnitude of the cross product that would be 1. But, as danago says, it would be simplest to use the dot product. $(2, -1, 1)\cdot(x, y, z)= 2x- y+ z= 0$.

That gives a single equation in three unknown values. If you really just want a vector perpendicular to (2, -1, 1), choose any values you want for, say, x and y, and solve for the corresponding z.

If you want to be able to express all vectors perpendicular to (2, -1, 1), solve for z as a function of x and y and use x and y as parameters.

## 1. How do you find a vector perpendicular to another vector?

To find a vector perpendicular to another vector, you can use the cross product. The cross product of two vectors will result in a vector that is perpendicular to both of the original vectors.

## 2. Can two vectors be perpendicular if they have the same magnitude?

No, two vectors can only be perpendicular if they have the same magnitude if they are also parallel. If two vectors have the same magnitude but are not parallel, they cannot be perpendicular to each other.

## 3. What is the formula for finding the cross product of two vectors?

The formula for finding the cross product of two vectors is given by:
A x B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k

## 4. Can a vector be perpendicular to itself?

No, a vector cannot be perpendicular to itself. In order for two vectors to be perpendicular, they must be pointing in different directions and have a 90 degree angle between them. A vector cannot have a 90 degree angle with itself.

## 5. How many vectors can be perpendicular to a given vector?

In three-dimensional space, there are an infinite number of vectors that can be perpendicular to a given vector. This is because for any given vector, there are an infinite number of planes that can contain it and have a perpendicular direction.

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