Question 1

Which of the following equations represents a plane in 3D space that is parallel to the $$xy-$$plane and passes through the point $$(2, -3, 5)$$?

Accepted Answer

$$z = 5$$

Question 2

Given two points $$P(1, 2, 3)$$ and $$Q(4, 6, 8)$$, what is the distance between them?

Accepted Answer

$$\sqrt{50} = 5\sqrt{2}$$

Question 3

Which of the following is the equation of a sphere with center at $$(1, -2, 3)$$ and radius $$4$$?

Accepted Answer

$$(x - 1)^2$ + (y + 2)^2$ + (z - 3)^2$ = 16$$

Question 4

A plane in space is parallel to the $$xz-$$plane and passes through the point $$(3, 7, -2). $$What is its equation?

Accepted Answer

$$y = 7$$

Question 5

If the equation of a sphere is $$x^2$$ + $$y^2$$ + $$z^2 - 4x + 6y - 8z = 0$, what are the coordinates of its center?

Accepted Answer

$$(2, -3, 4)$$

Question 6

Which of the following best describes the $$yz-$$plane in three-dimensional space?

Accepted Answer

$$x = 0$$

Question 7

A sphere has the equation $$(x - 2)^2$ + (y + 1)^2$ + (z - 5)^2$ = 9. $$What is its radius?

Accepted Answer

$$3$$

Question 8

Which of the following points lies in the $$xz-$$plane?

Accepted Answer

$$(4, 0, -2)$$

Question 9

What is the equation of the plane passing through the point $$(3, 11, 7)$$ and parallel to the $$yz-$$plane?

Accepted Answer

$$x = 3$$

Question 10

Given the points $$A(1, 2, 3)$$ and $$B(4, 6, 8)$$, what is the length of the segment $$AB$$?

Accepted Answer

$$\sqrt{50}$$

Question 11

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

Accepted Answer

$$y = c$$

Question 12

A sphere has center at $$(0, 0, 0)$$ and radius $$r. $$Which of the following is its equation?

Accepted Answer

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

Question 13

If a point $$P(x, y, z) is$$ $$5$$ units from the origin, which equation must it satisfy?

Accepted Answer

$$x^2$$ + $$y^2$$ + $$z^2 = 25$

Question 14

Which of the following is the equation of a plane parallel to the $$xy-$$plane and passing through $$(a, b, c)$$?

Accepted Answer

$$z = c$$

Question 15

Given the equation $$x^2$$ + $$y^2$$ + $$z^2 + 4x - 6y + 2z + 6 = 0$, what is the radius of the sphere?

Accepted Answer

$$5$$

Question 16

A medical imaging device locates a tumor at $$(2, -1, 5) in a $$patient's body. If the device can treat a spherical region of radius $$3$$ centered at this point, what is the equation of the treatment region?

Accepted Answer

$$(x - 2)^2$ + (y + 1)^2$ + (z - 5)^2$ = 9$$

Question 17

A drone flies from point $$A(0, 0, 0) to $$point $$B(3, 4, 12). $$What is the straight-line distance it travels?

Accepted Answer

$$13$$

Question 18

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

Accepted Answer

$$x = c$$

Question 19

A sphere has the equation $$x^2$$ + $$y^2$$ + $$z^2 - 2x + 4y - 6z = 11$. What is the center of the sphere?

Accepted Answer

$$(1, -2, 3)$$

Question 20

A chemical reaction occurs at all points that are $$2$$ units away from the origin. Which equation describes the set of all such points?

Accepted Answer

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

Section 2.1 - Three-Dimensional Coordinate Systems

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