Exploring Distance and Midpoint in 3D

The video tutorial introduces graphing in three dimensions, starting with the basics of the distance formula in two dimensions and extending it to three dimensions by adding a z-coordinate. It explains how to graph points in 3D space using x, y, and z axes, and provides a visual analogy of moving along the edges of a box. The tutorial also covers the distance formula in 3D, which is an extension of the Pythagorean theorem, and introduces the midpoint formula for three-dimensional points. The instructor plans to use computer tools to aid visualization and understanding.

What is the basic distance formula in two dimensions?

• X2 - X1 squared + Y2 - Y1 squared, then square root

• X2 - X1 squared + Y2 - Y1 squared

• X2 - X1 + Y2 - Y1

• X2 + X1 squared + Y2 + Y1 squared

In three dimensions, which additional variable is included in the distance formula?

• W

• Z

• T

• V

Which axes are used to graph points in three dimensions?

• X, Z, and W

• X, Y, and W

• Y, Z, and W

• X, Y, and Z

How do you graph the point (1, 2, 4) in three dimensions?

• Move 1 unit along X, 2 units up Y, and 4 units forward Z

• Move 4 units along X, 2 units up Y, and 1 unit forward Z

• Move 1 unit along X, 4 units up Y, and 2 units forward Z

• Move 2 units along X, 1 unit up Y, and 4 units forward Z

What shape can help visualize three-dimensional graphs?

• Box

• Sphere

• Cylinder

• Pyramid

What is the formula for the distance between two points in three dimensions?

• X2 - X1 squared + Y2 - Y1 squared + Z2 - Z1 squared, then square root

• X2 - X1 squared + Y2 - Y1 squared + Z2 - Z1 squared

• X2 - X1 squared + Y2 - Y1 squared

• X2 - X1 + Y2 - Y1 + Z2 - Z1

What is the midpoint formula in three dimensions?

• (X1 + X2) / 2, (Y1 + Y2) / 2, (Z1 - Z2) / 2

• (X1 + X2) / 2, (Y1 + Y2) / 2, (Z1 + Z2) / 2

• (X1 + X2) / 2, (Y1 + Y2) / 2

• (X1 - X2) / 2, (Y1 - Y2) / 2, (Z1 - Z2) / 2

What is the result of applying the distance formula to the points (1, 2, 4) and (3, 5, -1)?

• Square root of 35

• Square root of 45

• Square root of 55

• Square root of 50

How do you find the midpoint between (1, 2, 4) and (3, 5, -1)?

• (2, 3.5, 1.5)

• (2, 3.5, 1.5)

• (2, 3.5, 1.5)

• (2, 3.5, 1.5)

What theorem is the distance formula based on?

• Archimedes' Theorem

• Newton's Theorem

• Euclidean Theorem

• Pythagorean Theorem