# Calculus 3 : Vectors and Vector Operations

## Example Questions

### Example Question #11 : Vectors And Vector Operations

Find the angle between the two vectors.

No angle exists

Explanation:

To find the angle between two vector we use the following formula

and solve for .

Given

we find

Plugging these values in we get

To find  we calculate the  of both sides

and find that

### Example Question #12 : Angle Between Vectors

Find the approximate angle in degrees between the vectors .

Explanation:

We can compute the (acute) angle between the two vectors using the formula

Hence we have

### Example Question #13 : Angle Between Vectors

Find the angle in degrees between the vectors .

Explanation:

The correct answer is approximately  degrees.

To find the angle between two vectors, we use the equation .

Hence we have

(This answer is small due to the fact that the two vectors nearly point in the same direction, due to  and  being close in value.)

### Example Question #14 : Angle Between Vectors

Find the acute angle in degrees between the vectors .

Explanation:

To find the angle between two vectors, we use the formula

.

So we have

### Example Question #15 : Angle Between Vectors

Find the angle between the following vectors (to two decimal places):

1.46

1.62

1.18

2.89

0.98

1.46

Explanation:

The dot product is defined as:

Where theta is the angle between the two vectors.  Solving for theta:

To solve each component:

Putting it all together, we can solve for theta:

### Example Question #16 : Angle Between Vectors

Find the angle between vectors  and  and round to the nearest degree.

Explanation:

Write the formula to find the angle between two vectors.

Evaluate each term.

Substitute the values into the equation.

### Example Question #17 : Angle Between Vectors

Find the angle between the two vectors

Explanation:

In order to find the angle between the two vectors, we follow the formula

and solve for .

Using the vectors in the problem, we get

Simplifying we get

To solve for  we find the  of both sides and get

and find that

### Example Question #18 : Angle Between Vectors

What is the angle to the nearest degree between the vectors  and ?

Explanation:

In order to find the angle between the two vectors, we follow the formula

and solve for

Using the vectors in the problem, we get

Simplifying we get

To solve for

we find the

of both sides and get

and find that

### Example Question #19 : Angle Between Vectors

Find the angle between the two vectors

and

Round to the nearest degree.

Explanation:

In order to find the angle between the two vectors, we follow the formula

and solve for

Using the vectors in the problem, we get

Simplifying we get

To solve for

we find the

of both sides and get

and find that

### Example Question #20 : Angle Between Vectors

Find the angle between the two vectors

Explanation:

In order to find the angle between the two vectors, we follow the formula

and solve for .

Using the vectors in the problem, we get

Simplifying we get

To solve for  we find the  of both sides and get

and find that