Dot Product - AP Calculus BC

To evaluate the dot product of any two vectors, we multiply them component-wise.

.

All we need to do is multiply like components.

All we need to do is multiply the like components and add them together.

To find the dot product between two vectors

we calculate

so for

we have

To find the dot product between two vectors

we calculate

so for

we have

We can compute the length of a vector by taking the square root of the dot product of and , so the length of is:

We can compute the length of a vector by taking the square root of the dot product of and , so the length of is:

is not correct. This is saying effectively to add all the components of the two vectors together. The other two definitions are commonly used in computing angles between vectors and other objects, and can also be derived from each other.

This statement is true; it can be derived from the definition by setting the acute angle between the vectors to be ; the requirement for orthogonality. Additionally, if either vector has length , the vectors are still said to be orthogonal.

Let vector be represented as and vector be represented as .

The dot product of the vectors and is .

In this problem

Let vector be represented as and vector be represented as .

The dot product of the vectors and is .

In this problem

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 .

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.

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

The dot product for two vectors and

is defined as

Fo the given vectors

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 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 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 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: