### All Calculus 3 Resources

## Example Questions

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

Find the angle between the vectors and if , , and .

**Possible Answers:**

**Correct answer:**

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 : Vectors And Vector Operations

Two vectors **v** and **w **are separated by angle of 30^{o}. The vectors have the magnitudes:

What is the dot product of the two vectors .

**Possible Answers:**

**Correct answer:**

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 #43 : Vectors And Vector Operations

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

**Possible Answers:**

**Correct answer:**

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 #44 : Vectors And Vector Operations

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

**Possible Answers:**

**Correct answer:**

Next, we find the dot product

Plugging into the dot product formula, we get

. Solving for theta, we then get

### Example Question #45 : Vectors And Vector Operations

Find the angle to the nearest degree between the two vectors

**Possible Answers:**

**Correct answer:**

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 to the nearest degree between the two vectors

**Possible Answers:**

**Correct answer:**

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 #47 : Vectors And Vector Operations

Find the angle in degrees between the two vectors

**Possible Answers:**

**Correct answer:**

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 #48 : Vectors And Vector Operations

Find the angle between the vectors and , where and

Note: Use the dot product formula when finding the answer

**Possible Answers:**

**Correct answer:**

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

, and solving for theta, we get

### Example Question #49 : Vectors And Vector Operations

Determine the cosine of the angle between the following vectors:

**Possible Answers:**

**Correct answer:**

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 #50 : Vectors And Vector Operations

Find the angle between and in degrees

**Possible Answers:**

26

35

89

**Correct answer:**

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

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