Trigonometry

How to move a specific distance along a line determined by 2 points in 3d space!

Specific distance = m

Distance between the 2 points = D

Distance to move along line from Point #2 toward Point #1 = Displacement = m

Determine the coordinates of the point M (c, d, e), which is m units closer to Point#2

Given 2 points

Point #1 (a, b, c)

Point #2 (g. h, i)

1. Find the distance between the 2 points using Pythagorean Theorem

Think of moving from Point #1 to Point #2 by moving along the x-axis, then the y-axis, then the z-axis.

(g-a) = distance moved along the x-axis

(h-b) = distance moved along the y-axis

(i-c) = distance moved along the x-axisS

D = [(g-a)^2 + (h-e)^2 + (i-c)^2]^0.5

2. Determine the coordinates of the unit vector by dividing the distance moved along each axis by D.

Coordinates of unit vector = [(g-a) ÷ D], [(h-b) ÷ D], [(i-c) ÷ D]

x coordinate of unit vector = (g-a) ÷ D

y coordinate of unit vector = (h-b) ÷ D

z coordinate of unit vector = (i-c) ÷ D

Unit vector = [((g-a) ÷ D)^2 + ((h-e) ÷ D) ^2) + ((i-c) ÷ D) ^2]^0.5) = 1

If the value of the unit vector does not =1, go back and check your work.

3. Multiply each coordinate of the unit vector by m to determine the coordinates of the vector m. These coordinates will be added to coordinates of Point #1 to determine the coordinates of Point #3.

x coordinate of m vector = m * (g-a) ÷ D

y coordinate of m vector = m * (h-b) ÷ D

z coordinate of m vector = m * (i-c) ÷ D

4. To determine the coordinates of Point #3(d, e, f) that is m cm from Point #1 toward Point #2, add the coordinates of the m vector to the coordinates of Point #1.

d = x coordinate of Point #3 = a + (m * (g-a) ÷ D)

e = y coordinate of Point #3 = b + (m * (h-b) ÷ D)

f = z coordinate of Point #3 = c + (m * (i-c) ÷ D)

5. To determine the distance from Point #1 (a, b, c) to Point #3 (d, e, f), use Pythagorean Theorem

D = [(d-a)^2 + (e-b)^2 + (f-c)^2]^0.5

The answer should be m.

I wanted to move 2 cm from Point #1 toward Point #2, and I did.

Now let's see if this method works!! Point #1 = (2,3,1), Point #2 = (6,9,3)

I want to move 2 cm from Point #1 toward Point #2, that means m = 2 cm.

1. Find the distance between the 2 points using Pythagorean Theorem

D = [(g-a)^2 + (h-e)^2 + (i-c)^2]^0.5

D = [(6-2)^2 + (9-3)^2 + (3-1)^2]^0.5

D = [(4)^2 + (6)^2 + (2)^2]^0.5

D = [16 + 36 + (4)]^0.5

D = 56^0.5

D = 7.4833

So the line between these Point #1 and Point #2 is 7.483 units long

2. Determine the coordinates of the unit vector by dividing the distance moved along each axis by D.

Distance moved along x-axis = 4

Distance moved along y-axis = 6

Distance moved along z-axis = 2

x-coordinate of unit vector = 4 ÷ 7.4833 = 0.5345

y-coordinate of unit vector = 6 ÷ 7.483 = 0.8018

z-coordinate of unit vector = 2 ÷ 7.483 = 0.2673

Length of unit vector = [(0.5345)^2 + (0.8018)^2+ (0.2673)^2]^0.5 = 1

The length of the unit vector should = 1

3. Multiply each coordinate of the unit vector by m to determine the coordinates of the vector m.

x coordinate of m vector = m * (g-a) ÷ D = 2 * 0.5345 = 1.069

y coordinate of m vector = m * (h-b) ÷ D = 2 * 0.8018 = 1.6036

z coordinate of m vector = m * (i-c) ÷ D = 2 * 0.2673 = 0.5346

m vector = [1.069^2 + (1.6036)^2 + (1.5346)^2]^0.5 = 2

4. To determine the coordinates of the Point #3 (d, e, f) that is m cm from Point #1 toward Point #2, add the coordinates of the m vector to the coordinates of Point #1 (a, b, c).

Point #1 = (2, 3, 1)

x coordinate of Point #3 = 2 + 1.069 = 3.069

y coordinate of Point #3 = 3 +1.6036 = 4.6036

z coordinate of Point #3 = 1+ 0.5346 = 1.5346

Point #3 = (3.069, 4.6036, 1.5346)

5. To determine the distance from Point#1 to Point #3, use Pythagorean Theorem

D = [(d-a)^2 + (e-b)^2 + (f-c)^2]^0.5

D = [(3.069-2)^2 + (4.6036-3)^2 + (1.5346-1)^2]^0.5

D = [(1.069)^2 + (1.6036)^2 + (0.5346)^2]^0.5 = 2

D = 2 cm

I wanted to move 2 cm from Point #1 toward Point #2, and I did.

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