Question 1

Given the vector equation of a line $$\mathbf{r} = (1, -2, 3) + t\langle 2, 0, -1 \rangle$$, which of the following is the correct set of parametric equations for this line?

Accepted Answer

$$x = 1 + 2t,\ y = -2,\ z = 3 - t$$

Question 2

Which of the following equations represents a plane in $$\mathbb{R}^3$$ passing through the point $$(1,2,3)$$ with normal vector $$\langle 2, -1, 4 \rangle$$?

Accepted Answer

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

Question 3

What is the intersection between the line $$\mathbf{r} = (0, 1, 2) + t\langle 1, -1, 1 \rangle$$ and the plane $$x + y + z = 3$$?

Accepted Answer

The point $$(1, 0, 3)$$

Question 4

Which of the following is the general equation for a quadric surface?

Accepted Answer

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

Question 5

If the trace of the surface $$x^2$$ + $$y^2$$ - $$z^2 = 9$ in the xz$-plane is found, what type of conic section is it?

Accepted Answer

A hyperbola

Question 6

Which of the following surfaces has traces that are all circles in the $$xy$$, $$yz$$, and $$xz$$ planes?

Accepted Answer

Sphere

Question 7

Given the surface $$y^2$$ + $$2z^2$$ = $$3x^2$$, what is the trace in the plane $$z = 0$$?

Accepted Answer

$$y^2$$ = $$3x^2$$

Question 8

Which of the following best describes the surface $$x^2$$ + $$y^2$$ - $$z^2 = 1$?

Accepted Answer

A hyperboloid of one sheet

Question 9

Suppose you slice the surface $$z = x^2$ + y^2$ with the plane y = 1$. What is the resulting trace?

Accepted Answer

$$z = x^2$ + 1$$

Question 10

Which of the following equations represents a cylinder in $$\mathbb{R}^3$$?

Accepted Answer

$$x^2$$ + $$y^2 = 4$

Question 11

A plane in $$\mathbb{R}^3 is $$given by $$2x - 3y + z = 5. $$What is the normal vector to this plane?

Accepted Answer

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

Question 12

Which of the following is the equation of a hyperboloid of two sheets?

Accepted Answer

$$-x^2$$ - $$y^2$$ + $$z^2 = 1$

Question 13

If a surface has the equation $$z = x^2$ - y^2$, what is the trace in the plane x = 0$?

Accepted Answer

$$z = -y^2$

Question 14

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

Accepted Answer

Circles

Question 15

A line in $$\mathbb{R}^3 is $$given by $$\mathbf{r} = (3, -1, 2) + t\langle 1, 2, -1 \rangle. $$Which point lies on this line for $$t = -2$$?

Accepted Answer

$$(1, -5, 4)$$

Question 16

Which of the following is a possible real-world application of quadric surfaces?

Accepted Answer

Designing satellite dishes (paraboloids)

Question 17

Given the surface $$x^2$$ + $$y^2$$ + $$z^2 = 16$, what is the radius of the circle formed by the trace in the plane z = 0$?

Accepted Answer

4

Question 18

Which of the following surfaces is generated by revolving the parabola $$y = x^2$ about the y$-axis?

Accepted Answer

Paraboloid

Question 19

If the equation $$x^2$$ + $$y^2$$ - $$z^2 = 0$ is sliced by the plane z = 2$, what is the resulting trace?

Accepted Answer

$$x^2$$ + $$y^2 = 4$

Quadric Surfaces and Traces: Study Guide for Calculus Students

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