Question 1

Which of the following best describes the three-dimensional rectangular coordinate system?

Accepted Answer

It consists of three perpendicular axes and an origin at their intersection.

Question 2

What is the equation of a plane parallel to the $$xy-$$plane and passing through $$z = 5$$?

Accepted Answer

$$z = 5$$

Question 3

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

Accepted Answer

$$\sqrt{50}$$

Question 4

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

Accepted Answer

$$(x - 2)^2$ + (y + 1)^2$ + (z - 4)^2$ = 9$$

Question 5

Which of the following statements about the coordinate planes is correct?

Accepted Answer

The equation $$x = a$$ represents a plane parallel to the $$yz-$$plane.

Question 6

A drone flies from point $$A = (2, 3, 1) to $$point $$B = (5, 7, 4). $$What is the straight-line distance it travels?

Accepted Answer

$$\sqrt{34}$$

Question 7

Which of the following points lies on the sphere $$(x - 1)^2$ + (y + 2)^2$ + (z - 3)^2$ = 16$$?

Accepted Answer

$$(5, -2, 3)$$

Question 8

If a hospital is located at $$(0, 0, 0)$$ and an ambulance is at $$(6, 8, 0)$$, what is the distance between them?

Accepted Answer

$$10$$

Question 9

Which of the following is true about the right-hand rule in the three-dimensional coordinate system?

Accepted Answer

It determines the orientation of the axes.

Question 10

A sphere has the equation $$(x + 2)^2$ + (y - 3)^2$ + (z - 1)^2$ = 25. $$What is its center and radius?

Accepted Answer

Center $$(-2, 3, 1)$$, radius $$5$$

Question 11

Which equation represents a plane parallel to the $$xz-$$plane?

Accepted Answer

$$y = b$$

Question 12

A weather balloon is released from $$(0, 0, 0)$$ and rises to $$(3, 4, 12). $$What is the distance it has traveled from the origin?

Accepted Answer

$$13$$

Question 13

Which of the following is the correct formula for the distance between points $$P = (x_1$, y_1$, z_1$)$$ and $$Q = (x_2$, y_2$, z_2$) in$$ $$\mathbb{R}^3$$?

Accepted Answer

$$\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$ + (z_2$ - z_1$)^2$$}$

Question 14

A chemical reaction occurs at point $$(2, 5, 7) in a $$laboratory. If a sensor is placed at $$(2, 5, 2)$$, what is the distance between the reaction and the sensor?

Accepted Answer

$$5$$

Question 15

Which of the following equations represents a sphere centered at the origin with radius $$r$$?

Accepted Answer

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

Three-Dimensional Coordinate Systems (Calculus III Study Notes)

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