# Calculus 3 : Vectors and Vector Operations

## Example Questions

### Example Question #41 : Angle Between Vectors

Find the angle between the vectors  and  if , and .

Explanation:

Using the formula for the cross product between vectors  and , we have everything but theta. Plugging in what we were given, we get:

. Solving for , we get

### Example Question #42 : Angle Between Vectors

Two vectors v and are separated by angle of 30o.  The vectors have the magnitudes:

What is the dot product of the two vectors .

Explanation:

The angle theta between two vectors v and w is defined by:

Rearranging, we can solve for the dot product:

Substituting the given quantities:

### Example Question #41 : Vectors And Vector Operations

Find the angle between vectors  and . Use the dot product when finding the solution.

Explanation:

First, we must find the magnitude of the vectors.

Next, we find the dot product

Plugging into the dot product formula, we get

. Solving for theta, we then get

### Example Question #42 : Vectors And Vector Operations

Find the angle between vectors  and . Use the dot product when finding the solution.

Explanation:

Next, we find the dot product

Plugging into the dot product formula, we get

. Solving for theta, we then get

### Example Question #43 : Vectors And Vector Operations

Find the angle  to the nearest degree 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 #44 : Vectors And Vector Operations

Find the angle  to the nearest degree 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 #45 : Vectors And Vector Operations

Find the angle  in degrees 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 #46 : Vectors And Vector Operations

Find the angle between the vectors  and , where  and

Note: Use the dot product formula when finding the answer

Explanation:

To find the angle between the vectors, we use the formula for the dot product:

, and solving for theta, we get

### Example Question #47 : Vectors And Vector Operations

Determine the cosine of the angle between the following vectors:

Explanation:

The cosine of the angle, denoted by ,  between two vectors is given by the dot product of the vectors, which is the sum of the products of the corresponding components.

For our two vectors, the dot product is given by

### Example Question #48 : Vectors And Vector Operations

Find the angle between  and  in degrees

89

26

35

Explanation:

Step 1: Calculate

Step 2: Find the respective magnitudes of A and B

Step 3:

Use the formula to find the angle between two vectors  and .

Let the angle between the vectors be . Then