# Calculus 3 : Dot Product

## Example Questions

### Example Question #11 : Dot Product

For what angle(s) is the dot product of two vectors ?

Explanation:

We have the following equation that shows the relation between the dot product of two vectors, , to the relative angle between them ,

From this, we can see that the numerator  will be  whenever .

for all odd-multiples of , which in one rotation, includes .

### Example Question #12 : Dot Product

Compute

Explanation:

There is no correct way to compute the above. In order to take the dot product, the two vectors must have the same number of components. These vectors have and  components respectively.

### Example Question #13 : Dot Product

Compute

Explanation:

To computer the dot product, we multiply the values of common components together and sum their totals. The outcome is a scalar value, not a vector.

So we have

### Example Question #14 : 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 #15 : 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 #16 : 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 #17 : 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 #18 : 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 #19 : 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 #20 : Dot Product

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