Q: Which formula gives the angle $$\theta$$ between two planes with normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$?

$$\cos \theta = \dfrac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$$

Question 1

Given the planes $$x + y + z = 0$$ and $$2x - y + z = 0$$, what is the direction vector of their line of intersection?

Accepted Answer

$$\langle 2, -3, 1 \rangle$$

Question 2

Which formula gives the angle $$	heta$$ between two planes with normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$?

Accepted Answer

$$\cos 	heta = \dfrac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$$

Question 3

What is the distance between the parallel planes $$2x + y - z = 2$$ and $$2x + y - z = 8$$?

Accepted Answer

$$\dfrac{6}{\sqrt{6}}$$

Question 4

Which of the following pairs of planes are orthogonal?

Accepted Answer

$$2x - 3y + 2z = 3$$ and $$6x + 2y - 3z = 1$$

Question 5

Given the point $$P(1, 2, -1)$$ and the plane $$3x - 4y + 12z + 5 = 0$$, what is the distance from $$P to $$the plane?

Accepted Answer

$$\dfrac{|3(1) - 4(2) + 12(-1) + 5|}{\sqrt{3^2$ + (-4)^2$ + 12^2$}}$$

Question 6

Which equation represents a right circular cylinder of radius 3 centered on the $$z-$$axis?

Accepted Answer

$$x^2$$ + $$y^2 = 9$

Question 7

Which of the following is the definition of a cylindrical surface?

Accepted Answer

A set of lines parallel to a given line passing through a given curve.

Question 8

Which quadric surface is represented by the equation $$\dfrac{x^2$}{a^2$} + \dfrac{y^2$}{b^2$} + \dfrac{z^2$}{c^2$} = 1$$?

Accepted Answer

Ellipsoid

Question 9

Which surface is described by the equation $$\dfrac{x^2$}{a^2$} - \dfrac{y^2$}{b^2$} - \dfrac{z^2$}{c^2$} = 1$$?

Accepted Answer

Hyperboloid of two sheets

Question 10

Which of the following equations represents an elliptic paraboloid?

Accepted Answer

$$z = \dfrac{x^2$}{a^2$} + \dfrac{y^2$}{b^2$}$$

Question 11

A plane has normal vector $$\vec{n}_1 = \langle 1, 2, 3 \rangle$$ and another has $$\vec{n}_2 = \langle 2, 4, 6 \rangle. $$What is the relationship between the planes?

Accepted Answer

They are parallel.

Question 12

Which of the following is a symmetric equation for the line of intersection of the planes $$x + y + z = 0$$ and $$2x - y + z = 0$$?

Accepted Answer

$$\dfrac{x}{2} = \dfrac{y}{-3} = \dfrac{z}{1}$$

Question 13

Which surface is represented by $$z = \dfrac{x^2$}{a^2$} - \dfrac{y^2$}{b^2$}$$?

Accepted Answer

Hyperbolic paraboloid

Question 14

If a plane has equation $$ax + by + cz + d = 0$$, what is the magnitude of its normal vector?

Accepted Answer

$$\sqrt{a^2$ + b^2$ + c^2$}$$

Question 15

Which of the following equations represents a hyperboloid of one sheet?

Accepted Answer

$$\dfrac{x^2$}{a^2$} + \dfrac{y^2$}{b^2$} - \dfrac{z^2$}{c^2$} = 1$$

Question 16

A set of lines parallel to a given line passing through a given curve forms which type of surface?

Accepted Answer

Cylindrical surface

Question 17

Which of the following is NOT a quadric surface?

Accepted Answer

Cylinder

Question 18

Given the equation $$x^2$$ + $$z^2 = 25$, what type of surface does this represent?

Accepted Answer

A cylinder parallel to the $$y-$$axis

Question 19

Which of the following best describes the traces of an ellipsoid in the coordinate planes?

Accepted Answer

Ellipses in all three planes

Planes, Cylinders, and Quadric Surfaces in Physics with Calculus

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