# Calculus 3 : Dot Product

## Example Questions

### Example Question #61 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors  and

The dot product can be found following the example above:

### Example Question #62 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors  and

The dot product can be found following the example above:

### Example Question #63 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors  and

The dot product can be found following the example above:

### Example Question #64 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors  and

The dot product can be found following the example above:

### Example Question #65 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors  and

The dot product can be found following the example above:

Observe from this zero result that the two vectors must be perpendicular!

### Example Question #66 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors

and

The dot product can be found following the example above:

### Example Question #67 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors

and

The dot product can be found following the example above:

### Example Question #68 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors

and

The dot product can be found following the example above:

### Example Question #69 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.

Explanation:

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors

and

The dot product can be found following the example above:

### Example Question #70 : Dot Product

Given the following two vectors,  and , calculate the dot product between them,.