Understanding the Distance Formula in 3D

The video tutorial explains how to find the distance between two points in three-dimensional space using the distance formula. It begins with an introduction to the formula, followed by a step-by-step example calculation. The tutorial then visualizes the points in 3D space to provide a better understanding of the concept. It also demonstrates a visual method to calculate the distance and concludes with a derivation of the distance formula using geometric principles.

What is the primary purpose of the distance formula in three-dimensional space?

• To calculate the area of a triangle

• To find the distance between two points

• To determine the volume of a cube

• To measure the angle between two lines

Which of the following is the correct expression for the distance formula in 3D?

• d = (x2 - x1) + (y2 - y1) + (z2 - z1)

• d = (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2

• d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

• d = √((x2 - x1) + (y2 - y1) + (z2 - z1))

In the example calculation, what is the distance between the points (1, 2, -5) and (4, 6, 7)?

• 13

• 15

• 12

• 10

When plotting points in 3D space, which axis is typically represented vertically?

• Y-axis

• None of the above

• X-axis

• Z-axis

How many units are traveled along the Z-axis from point (1, 2, -5) to (4, 6, 7)?

• 10 units

• 12 units

• 14 units

• 16 units

What geometric shape is used to visually understand the distance between two points in 3D?

• Square

• Circle

• Triangle

• Parallelogram

In the geometric approach, what is the hypotenuse of the right triangle formed by sides 3 and 4?

• 6

• 8

• 7

• 5

What is the relationship between the sides of a right triangle in the context of the distance formula?

• a^2 + b^2 = c^2

• a^2 + b^2 = c

• a + b = c

• a^2 - b^2 = c^2

In the derivation using variables, what does 'd' represent?

• The difference between x-values

• The distance between two points

• The sum of y-values

• The product of z-values

What is the final form of the distance formula derived using variables?

• d = x + y + z

• d = √(x^2 + y^2 + z^2)

• d = x^2 + y^2 + z^2

• d = √(x + y + z)