# Calculus 3 : Dot Product

## Example Questions

### Example Question #31 : 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 #32 : 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 #33 : 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:

Note that if you were to find the magnitude of each vector and multiply the two, you'd find the result. This is because these two vectors happen to be parallel.

### Example Question #34 : 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:

The zero result shows that these two vectors are perpendicular.

### Example Question #35 : 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:

Note that since we found a zero value, these two vectors must be perpendicular!

### Example Question #36 : 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 #37 : 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 #38 : Dot Product

Find the dot product between the two vectors.

Explanation:

The dot product for two vectors  and

is defined as

Fo the given vectors

### Example Question #39 : Dot Product

Find the dot product of the two vectors.

Explanation:

The dot product for two vectors  and

is defined as

Fo the given vectors

### Example Question #40 : Dot Product

Find the dot product  of the two vectors