# Calculus 3 : Dot Product

## Example Questions

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### Example Question #1 : Dot Product

Evaluate the dot product between , and .

Explanation:

All we need to do is multiply like components.

### Example Question #2 : Dot Product

Evaluate the dot product of , and .

Explanation:

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

### Example Question #3 : Dot Product

Find the dot product of the following vectors:

Explanation:

To find the dot product between two vectors

we calculate

so for

we have

### Example Question #4 : Dot Product

What is the length of the vector

?

Explanation:

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

### Example Question #5 : Dot Product

Find the dot product of the following vectors:

Explanation:

To find the dot product between two vectors

we calculate

so for

we have

### Example Question #6 : Dot Product

What is the length of the vector

?

Explanation:

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

### Example Question #7 : Dot Product

Which of the following cannot be used as a definition of the dot product of two real-valued vectors?

They may all be used

, where  is the angle between .

Explanation:

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.

### Example Question #8 : Dot Product

Which of the following is true concerning the dot product of two vectors?

is well-defined as long as each vector is the same dimension

None of the other statements are true.

The dot product of two vectors is never negative.

if and only if  are orthogonal.

The dot product of two vectors is never a scalar.

if and only if  are orthogonal.

Explanation:

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.

### Example Question #9 : Dot Product

What is the dot product of vectors  and ?

Explanation:

Let vector  be represented as   and vector   be represented as  .

The dot product of the vectors   and  is .

In this problem

### Example Question #10 : Dot Product

What is the dot product of vectors  and ?

Does not exist