### All Calculus 3 Resources

## Example Questions

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

Find the angle between the two vectors.

**Possible Answers:**

No angle exists

**Correct answer:**

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 .

**Possible Answers:**

None of the other answers

**Correct answer:**

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 .

**Possible Answers:**

None of the other answers

**Correct answer:**

None of the other answers

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 .

**Possible Answers:**

None of the other answers

**Correct answer:**

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):

**Possible Answers:**

1.46

1.62

1.18

2.89

0.98

**Correct answer:**

1.46

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.

**Possible Answers:**

**Correct answer:**

Write the formula to find the angle between two vectors.

Evaluate each term.

Substitute the values into the equation.

The correct answer is:

### Example Question #17 : Angle Between Vectors

Find the angle 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 #18 : Angle Between Vectors

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

**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 #19 : Angle Between Vectors

Find the angle between the two vectors

and

Round to the nearest degree.

**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 #20 : Angle Between Vectors

Find the angle 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

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