Calculus 3 : Angle between Vectors

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Angle Between Vectors

Find the angle between these two vectors, , and .

Possible Answers:

Correct answer:

Explanation:

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:

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 ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 : Vectors And Vector Operations

What is the angle between the vectors  and ?

Possible Answers:

Correct answer:

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 #5 : Angle Between Vectors

What is the angle between the vectors  and ?

Possible Answers:

Correct answer:

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 #6 : Angle Between Vectors

Find the approximate acute angle in degrees between the vectors .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

To find the angle between two vectors, use the formula

.

Example Question #7 : Angle Between Vectors

Find the angle between the following two vectors.

Possible Answers:

Correct answer:

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 #8 : Angle Between Vectors

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

Possible Answers:

Correct answer:

Explanation:

To find the angle between vectors, we use the formula

.

Substituting in our values, we get

Example Question #9 : Angle Between Vectors

Find the angle between the two vectors. 


Possible Answers:

No angle exists

Correct answer:

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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