Calculus 3 : Normal Vectors

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #81 : Normal Vectors

Find the normal vector of the plane that is parallel to the plane given by the equation 

Possible Answers:

Correct answer:

Explanation:

To solve the problem, we use the fact that two parallel planes have the same normal vector. The equation of on of the planes is given, and from that we know its normal vector is , which is the normal vector of the plane in question

Example Question #81 : Normal Vectors

Find the normal vector to the plane given the vectors on the plane

 and 

 

Possible Answers:

Correct answer:

Explanation:

To find the normal vector to the plane containing vectors  and , we find the determinant of the 3x3 matrix 

Plugging in the vectors and solving, we get

Example Question #83 : Normal Vectors

Find the normal vector to the plane given the vectors on the plane

 and 

Possible Answers:

Correct answer:

Explanation:

To find the normal vector to the plane containing vectors  and , we find the determinant of the 3x3 matrix 

Plugging in the vectors and solving, we get

Example Question #84 : Normal Vectors

Find the vector normal to the plane given by the following vectors:

Possible Answers:

Correct answer:

Explanation:

First, we must write the determinant in order to take the cross product of the two vectors:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

The difference between zero and the zero vector is an important one, because the result of a cross product is always a vector (the dot product of two vectors gives a scalar). 

Example Question #85 : Normal Vectors

Find the normal vector to the plane containing the vectors  and 

Possible Answers:

Correct answer:

Explanation:

To find the normal vector to the plane containing vectors  and , we take the cross product of the two. 

To find the cross product between the vectors  and , we find the determinant of the 3x3 matrix  which follows the formula 

Applying to the vectors from the problem statement, we get

Example Question #86 : Normal Vectors

Find the normal vector to the plane containing the vectors  and 

Possible Answers:

Correct answer:

Explanation:

To find the normal vector to the plane containing vectors  and , we take the cross product of the two. 

To find the cross product between the vectors  and , we find the determinant of the 3x3 matrix  which follows the formula 

Applying to the vectors from the problem statement, we get

Example Question #87 : Normal Vectors

Find the normal vector to the following vectors:

Possible Answers:

Correct answer:

Explanation:

To find the normal vector, we must take the cross product of the two vectors.

Now, we can write the determinant in order to take the cross product of the two vectors:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

Example Question #88 : Normal Vectors

Determine the vector normal to the plane created by the following two vectors:

Possible Answers:

Correct answer:

Explanation:

The normal vector to a plane is given by the cross product of two vectors in that plane. 

So, we write the determinant in order to take the cross product of the two vectors:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

Example Question #89 : Normal Vectors

Find the normal vector to the plane containing the vectors  and 

Possible Answers:

Correct answer:

Explanation:

To find the normal vector to a plane containing vectors  and , you take the cross product of the two vectors. 

To find the cross product between the two vectors  and , you take the determinant of the 3x3 matrix 

Using the vectors from the problem statement, we get

Example Question #90 : Normal Vectors

Find the normal vector to the plane containing the vectors  and 

Possible Answers:

Correct answer:

Explanation:

To find the normal vector to a plane containing vectors  and , you take the cross product of the two vectors. 

To find the cross product between the two vectors  and , you take the determinant of the 3x3 matrix 

Using the vectors from the problem statement, we get

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