# Calculus 3 : Normal Vectors

## Example Questions

### Example Question #31 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

The two vectors are not orthogonal.

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are not orthogonal.

### Example Question #32 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

The two vectors are not orthogonal.

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are not orthogonal.

### Example Question #33 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

The two vectors are not orthogonal.

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are orthogonal.

### Example Question #34 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

The two vectors are not orthogonal.

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are not orthogonal.

### Example Question #35 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are not orthogonal.

The two vectors are orthogonal.

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are orthogonal.

### Example Question #36 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are not orthogonal.

The two vectors are orthogonal.

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are not orthogonal.

### Example Question #37 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

The two vectors are not orthogonal.

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are orthogonal.

### Example Question #38 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

The two vectors are not orthogonal.

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are not orthogonal.

### Example Question #39 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are orthogonal.

It cannot be determined unless  is known.

The two vectors are not orthogonal.

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are not orthogonal.

### Example Question #40 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

The two vectors are not orthogonal.

The two vectors are orthogonal.

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and

The two vectors are orthogonal.