### All Calculus 3 Resources

## Example Questions

### Example Question #1 : Angle Between Vectors

Find the angle between these two vectors, , and .

**Possible Answers:**

**Correct answer:**

Lets remember the formula for finding the angle between two vectors.

### Example Question #2 : Angle Between Vectors

Calculate the angle between , .

**Possible Answers:**

**Correct answer:**

Lets recall the equation for finding the angle between vectors.

### Example Question #3 : Angle Between Vectors

What is the angle between the vectors and ?

**Possible Answers:**

**Correct answer:**

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

where is the dot product of the vectors and , respectively.

and are the magnitudes of vectors and , respectively.

is the angle between the two vectors.

Let vector be represented as and vector be represented as .

The dot product of the vectors and is .

The magnitude of vector is and vector is .

Rearranging the dot product formula to solve for gives us

For this problem,

### Example Question #4 : Vectors And Vector Operations

What is the angle between the vectors and ?

**Possible Answers:**

**Correct answer:**

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

where is the dot product of the vectors and , respectively.

and are the magnitudes of vectors and , respectively.

is the angle between the two vectors.

Let vector be represented as and vector be represented as .

The dot product of the vectors and is .

The magnitude of vector is and vector is .

Rearranging the dot product formula to solve for gives us

For this problem,

The vectors are perpendicular

### Example Question #5 : Vectors And Vector Operations

What is the angle between the vectors and ?

**Possible Answers:**

**Correct answer:**

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

where is the dot product of the vectors and , respectively.

and are the magnitudes of vectors and , respectively.

is the angle between the two vectors.

Let vector be represented as and vector be represented as .

The dot product of the vectors and is .

The magnitude of vector is and vector is .

Rearranging the dot product formula to solve for gives us

For this problem,

### Example Question #4 : Angle Between Vectors

What is the angle between the vectors and ?

**Possible Answers:**

**Correct answer:**

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

where is the dot product of the vectors and , respectively.

and are the magnitudes of vectors and , respectively.

is the angle between the two vectors.

Let vector be represented as and vector be represented as .

The dot product of the vectors and is .

The magnitude of vector is and vector is .

Rearranging the dot product formula to solve for gives us

For this problem,

The two vectors are parallel.

### Example Question #5 : Angle Between Vectors

Find the approximate acute angle in degrees between the vectors .

**Possible Answers:**

None of the other answers

**Correct answer:**

To find the angle between two vectors, use the formula

.

### Example Question #6 : Angle Between Vectors

Find the angle between the following two vectors.

**Possible Answers:**

**Correct answer:**

In order to find the angle between two vectors, we need to take the quotient of their dot product and their magnitudes:

Therefore, we find that

.

### Example Question #7 : Angle Between Vectors

Find the (acute) angle between the vectors in degrees.

**Possible Answers:**

**Correct answer:**

To find the angle between vectors, we use the formula

.

Substituting in our values, we get

### Example Question #8 : Angle Between Vectors

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