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Calculus 3 : Calculus 3

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Example Questions

Example Question #151 : Calculus 3

Find the equation of the plane containing the point $\left ( 4,0,2\right )$ , and is parallel to the plane with the equation 5x+2y+2z=15

Possible Answers:

5x+2y+z=24

5x+2y+2z=16

5x+2y+2z=21

5x+2y+2z=24

Correct answer:

5x+2y+2z=24

Explanation:

We were given a point on the plane, and we need the normal vector to the plane. It is known that two planes that are parallel to each other have the same normal vector, so in this case $n=\left \langle 5,2,2\right \rangle$ (given by the equation of the other plane). To complete the problem, we use the equation A(x-x_0)+B(y-y_0)+C(z-z_0)=0 , where $n=\left \langle A,B,C\right \rangle$ and the point on the plane is (x_0,y_0,z_0) . Using the information we have, we get:

5(x-4)+2(y-0)+2(z-2)=0 . Through algebraic manipulation, we then get:

5x+2y+2z=24

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Example Question #32 : Equations Of Lines And Planes

Find the equation of the plane that contains the point (5,-1,2) and is parallel to the plane 2x+y+3z=6

Possible Answers:

2x+y+3z=12

2x+y+3z=15

2x+y=11

2x+y-3z=15

Correct answer:

2x+y+3z=15

Explanation:

To solve the problem, we will use the formula for the equation of a plane with a normal vector $n=\left \langle A,B,C\right \rangle$ and a point (x_0,y_0,z_0) :

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

We have the point, which from the problem statement is (5,-1,2) .

The normal vector is given by the equation of the plane that is in parallel to the one we are forming the equation for. That is $n=\left \langle 2,1,3\right \rangle$ .

Putting everything we know into the formula and solving, we get

2(x-5)+1(y+1)+3(z-2)=0

Simplifying, we get

2x+y+3z=15

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Example Question #33 : Equations Of Lines And Planes

Find the equation of the plane that contains the point (6,11,10) and is parallel to the plane 3x+-y+2z=5

Possible Answers:

x-6y+2z=15

3x-y+2z=49

3x+y+2z=49

x-y+5z=9

Correct answer:

3x-y+2z=49

Explanation:

To solve the problem, we will use the formula for the equation of a plane with a normal vector $n=\left \langle A,B,C\right \rangle$ and a point (x_0,y_0,z_0) :

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

We have the point, which from the problem statement is (6,11,10) .

The normal vector is given by the equation of the plane that is in parallel to the one we are forming the equation for. That is $n=\left \langle 3,-1,2\right \rangle$ .

Putting everything we know into the formula and solving, we get

3(x-6)-1(y+11)+2(z-10)=0

Simplifying, we get

3x-y+2z=49

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Example Question #34 : Equations Of Lines And Planes

Find the equation of the plane given by a point on the plane (-1,3,-4) and the normal vector to the plane $n=\left \langle 6,7,1\right \rangle$

Possible Answers:

6x+7y-z=21

6x+7y+z=11

6x-7y+z=11

6x+7y-z=-11

Correct answer:

6x+7y+z=11

Explanation:

To find the equation of a plane, we use the normal vector $n=\left \langle A,B,C\right \rangle$ and a point on the plane (x_0,y_0,z_0) . Using this information, we use the formula for a plane:

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

Using the information from the problem statement, we then get

6(x+1)+7(y-3)+1(z+4)=0

Rearranging through algebra, we get:

6x+7y+z=11

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Example Question #35 : Equations Of Lines And Planes

Find the equation of the plane that contains the point (3,-1,4) and is parallel to the plane 4x-8y+z=22

Possible Answers:

4x-8y+z=24

4x+8y+z=12

4x-8y+z=15

4x-8y+z=20

Correct answer:

4x-8y+z=24

Explanation:

To find the equation of a plane, we use the normal vector $n=\left \langle A,B,C\right \rangle$ and a point on the plane (x_0,y_0,z_0) . Using this information, we use the formula for a plane:

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

To find the normal vector, it is known that two parallel planes have the same normal vector. Using this and the point on the plane, we then get

4(x-3)+-8(y+1)+1(z-4)=0

Rearranging through algebra, we get:

4x-8y+z=24

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Example Question #41 : 3 Dimensional Space

Find the equation of the plane given by the normal vector $n=\left \langle 7,5,2\right \rangle$ and a point on the plane $\left ( 1,3,5\right )$

Possible Answers:

7x+5y+2z=22

7x+5y-4z=10

7x+5y+2z=32

Correct answer:

7x+5y+2z=32

Explanation:

To find the equation of the plane with a normal vector $n=\left \langle A,B,C\right \rangle$ and a point (x_0,y_0,z_0) , we use the formula

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

Using the information from the problem statement, we get

7(x-1)+5(y-3)+2(z-5)=0

This simplifies to 7x+5y+2z=32

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Example Question #41 : Equations Of Lines And Planes

Find the equation of the plane given by the normal vector $n=\left \langle 2,-5,6\right \rangle$ and a point on the plane $\left ( 3,7,9\right )$

Possible Answers:

2x-5y+6z=16

2x+5y+6z=25

2x-5y+6z=25

2x-5y+6z=24

Correct answer:

2x-5y+6z=25

Explanation:

To find the equation of the plane with a normal vector $n=\left \langle A,B,C\right \rangle$ and a point (x_0,y_0,z_0) , we use the formula

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

Using the information from the problem statement, we get

2(x-3)-5(y-7)+6(z-9)=0

This simplifies to 2x-5y+6z=25

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Example Question #151 : Calculus 3

Find the equation of the plane that contains the point $\left ( 3,-6,7\right )$ and has a normal vector $n=\left \langle 4,2,1\right \rangle$

Possible Answers:

3x+2y+z=7

4x+2y+z=7

4x+2y+z=14

4x+2y-z=7

Correct answer:

4x+2y+z=7

Explanation:

To find the equation of a plane with a point (x_0,y_0,z_0) and a normal vector $n=\left \langle A,B,C\right \rangle$ , we use the formula

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

Using the information from the problem statement, we get

4(x-3)+2(y+6)+1(z-7)=0

Simplifying, we then get

4x+2y+z=7

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Example Question #44 : 3 Dimensional Space

Find the equation of the plane that contains the point (5,-1,2) and contains the normal vector $\left \langle 4,5,5\right \rangle$

Possible Answers:

4x+5y+5z=20

2x+5y+5z=25

4x+5y+5z=25

4x+y+3z=25

Correct answer:

4x+5y+5z=25

Explanation:

To find the equation of a plane with a point (x_0,y_0,z_0) and a normal vector $n=\left \langle A,B,C\right \rangle$ , we use the formula

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

Using the information from the problem statement, we get

4(x-5)+5(y+1)+5(z-2)=0

Simplifying, we then get

4x+5y+5z=25

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Example Question #45 : 3 Dimensional Space

Find the equation of the plane containing these points:

A=(1, 2, 3) , B=(0, 3, -6) , C=(-2, 4, 2)

Possible Answers:

x+10y+z=-24

x+2y+3z=24

x+10y+z=0

x+10y+z=24

Correct answer:

x+10y+z=24

Explanation:

To find the equation of a plane, we need the normal vector and a point on the plane. The normal vector is found by taking the cross product of two vectors on the plane.

To find two vectors, simply find the difference between terminal and initial points:

$\vec{AB}=\left \langle -1, 1, -9\right \rangle$ , $\vec{BC}=\left \langle -2, 1, -8 \right \rangle$

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

$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ -1&1 &-9 \\ -2&1 &-8 \end{vmatrix}$

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:

$-8\hat{i}-\hat{k}+18\hat{j}-(-2\hat{k}-9\hat{i}+8\hat{j})=\hat{i}+10\hat{j}+\hat{k}=\left \langle 1, 10, 1\right \rangle$

Plugging this into the formula

a(x-x_0)+b(y-y_0)+c(z-z_0)=0 , where $\vec{n}=\left \langle a, b, c\right \rangle$ and (x_0, y_0, z_0) is any of the given points (for example (1, 2 ,3) ), we get

1(x-1)+10(y-2)+1(z-3)=0

which simplified becomes

x+10y+z=24

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #151 : Calculus 3

Example Question #32 : Equations Of Lines And Planes

Example Question #33 : Equations Of Lines And Planes

Example Question #34 : Equations Of Lines And Planes

Example Question #35 : Equations Of Lines And Planes

Example Question #41 : 3 Dimensional Space

Example Question #41 : Equations Of Lines And Planes

Example Question #151 : Calculus 3

Example Question #44 : 3 Dimensional Space

Example Question #45 : 3 Dimensional Space

All Calculus 3 Resources

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