# Calculus 3 : Angle between Vectors

## Example Questions

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### Example Question #1 : Angle Between Vectors

Find the angle between these two vectors, , and .

Explanation:

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

### Example Question #2 : Angle Between Vectors

Calculate the angle between .

Explanation:

Lets recall the equation for finding the angle between vectors.

### Example Question #3 : Angle Between Vectors

What is the angle between the vectors  and ?

Explanation:

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 ?

Explanation:

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 : Angle Between Vectors

What is the angle between the vectors  and ?

Explanation:

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 #6 : Angle Between Vectors

What is the angle between the vectors  and ?

Explanation:

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 #7 : Angle Between Vectors

Find the approximate acute angle in degrees between the vectors .

Explanation:

To find the angle between two vectors, use the formula

.

### Example Question #8 : Angle Between Vectors

Find the angle between the following two vectors.

Explanation:

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 #9 : Angle Between Vectors

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

Explanation:

To find the angle between vectors, we use the formula

.

Substituting in our values, we get

### Example Question #10 : Angle Between Vectors

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

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