# Calculus 3 : Dot Product

## Example Questions

### Example Question #91 : Dot Product

Find the length  of the vector .

Explanation:

To find the length  of the vector , we take the square root of the dot product :

### Example Question #92 : Dot Product

Find the length  of the vector .

Explanation:

To find the length  of the vector , we take the square root of the dot product :

### Example Question #93 : Dot Product

Find the length  of the vector .

Explanation:

To find the length  of the vector , we take the square root of the dot product :

### Example Question #94 : Dot Product

Find the dot product of the two vectors

Explanation:

The dot product for the vectors

is defined as

For the vectors in this problem we find that

### Example Question #95 : Dot Product

Find the dot product of the two vectors

Explanation:

The dot product for the vectors

is defined as

For the vectors in this problem we find that

### Example Question #411 : Vectors And Vector Operations

Calculate the dot product

Explanation:

The dot product for the vectors

is defined as

For the vectors in this problem we find that

### Example Question #412 : Vectors And Vector Operations

Find the vector projection of  onto .

Explanation:

To find the vector projection of  onto , we take the scalar projection , and multiply it to a vector of unit length; .

Hence for the vector projection of  onto , we have

.

### Example Question #413 : Vectors And Vector Operations

Find the vector projection of  onto .

Explanation:

To find the vector projection of  onto , we take the scalar projection , and multiply it to a vector of unit length; .

Hence for the vector projection of  onto , we have

.

### Example Question #414 : Vectors And Vector Operations

Find the dot product of  and

Explanation:

Simplify: