Question 1

Which of the following is the general equation for a plane in space passing through the point $$P_0$$($$x_0$$, $$y_0$$, $$z_0$$)$$ with normal vector $$\vec{n} = \langle a, b, c \rangle$$?

Accepted Answer

$$a(x - x_0$) + b(y - y_0$) + c(z - z_0$) = 0$$

Question 2

Given the plane $$2x - 3y + 6z = 12$$, what is a normal vector to this plane?

Accepted Answer

$$\langle 2, -3, 6 \rangle$$

Question 3

Which of the following equations represents a plane parallel to the $$yz-$$plane?

Accepted Answer

$$x = a$$

Question 4

Suppose you have two planes with normal vectors $$\vec{n}_1 = \langle 2, -1, 3 \rangle$$ and $$\vec{n}_2 = \langle 4, -2, 6 \rangle. $$What is the relationship between the planes?

Accepted Answer

The planes are parallel.

Question 5

Which of the following surfaces is represented by the equation $$\frac{x^2$}{9} + \frac{y^2$}{16} + \frac{z^2$}{25} = 1$$?

Accepted Answer

An ellipsoid

Question 6

Find the $$x-$$intercept of the plane $$2x - 3y + 6z = 12.$$

Accepted Answer

$$(6, 0, 0)$$

Question 7

Which of the following best describes the set of points where the plane $$2x - 3y + 6z = 12$$ intersects the $$yz-$$plane?

Accepted Answer

$$x = 0$$, so $$-3y + 6z = 12$$

Question 8

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

Accepted Answer

$$\frac{x^2$}{4} + \frac{y^2$}{9} - \frac{z^2$}{16} = 1$$

Question 9

Given the points $$A(1,1,2)$$, $$B(0,4,1)$$, and $$C(-2,1,4)$$, which of the following is a correct first step to find the equation of the plane containing these points?

Accepted Answer

Find two vectors in the plane, such as $$\overrightarrow{AB}$$ and $$\overrightarrow{AC}.$$

Question 10

Which of the following is the equation of a plane passing through $$(-2, 4, 1)$$ and parallel to the plane $$3x - 2y + 4z = 5$$?

Accepted Answer

$$3x - 2y + 4z = d$$, where $$d = 3(-2) - 2(4) + 4(1)$$

Question 11

Which of the following is a necessary and sufficient condition for two planes to be orthogonal?

Accepted Answer

The dot product of their normal vectors is zero.

Question 12

Which of the following equations represents a cone?

Accepted Answer

$$z^2$$ = $$x^2$$ + $$y^2$$

Question 13

A surface is given by $$z = y^2$. Which of the following best describes its traces in the xz$-plane?

Accepted Answer

A parabola opening upwards

Question 14

Which of the following is the general form of a quadratic surface?

Accepted Answer

$$Ax^2$$ + $$By^2$$ + $$Cz^2 + $$Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$$

Question 15

Which of the following surfaces is represented by the equation $$\frac{y^2$}{4} + \frac{z^2$}{4} = x$$?

Accepted Answer

An elliptic paraboloid

Question 16

Which of the following best describes the traces of the surface $$x^2$$ + $$y^2 = z$ in the xy$-plane?

Accepted Answer

Circles for each fixed $$z$$

Question 17

Which of the following equations represents a hyperbolic paraboloid?

Accepted Answer

$$z = x^2$ - y^2$

Question 18

If a plane has the equation $$ax + by + cz = d$$, what is the geometric meaning of the coefficients $$a$$, $$b$$, and $$c$$?

Accepted Answer

They are the components of the normal vector to the plane.

Planes and Quadratic Surfaces in Three-Dimensional Space

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