Find the point 4/5 of the way from (7, −10, −4) to (7, 0, 10).
Not sure where to go about with this. I'm assuming I find the distance from the two points first..
you don't need the distance
just go 4/5 in each direction
Do you mind elaborating on that, I don't really understand..
Just set up the ratios.
Let the point be (x,y,z)
I will do the y coordinate, you do the others the same way.
(y - (-10) )/(0 - y) = 4/5
(y+10)/-y = 4/5
5y + 50 = -4y
9y = -50
y = -50/9
or , using vectors (I am using <......> to represent the vectors
<x,y,z> = (5/9)<7,0,10> + (4/9)<7,0,10>
= ....
To find the point that is 4/5 of the way from (7, -10, -4) to (7, 0, 10), you can follow these steps:
1. Find the distance between the two points using the distance formula.
The distance formula between two points in 3D space is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
In this case, the coordinates of the two points are:
Point A: (7, -10, -4)
Point B: (7, 0, 10)
Calculate the differences in each coordinate:
Δx = 7 - 7 = 0
Δy = 0 - (-10) = 10
Δz = 10 - (-4) = 14
Calculate the square of each difference:
(Δx)^2 = 0^2 = 0
(Δy)^2 = 10^2 = 100
(Δz)^2 = 14^2 = 196
Add up the squares of the differences:
d = sqrt(0 + 100 + 196) = sqrt(296) ≈ 17.20
So the distance between the two points is approximately 17.20.
2. Determine 4/5 of the distance.
To find the point that is 4/5 of the way from Point A to Point B, you need to multiply the distance by 4/5:
4/5 of 17.20 = (4/5) * 17.20 = 13.76
So, 4/5 of the distance is approximately 13.76.
3. Calculate the coordinates of the point.
To find the coordinates of the point that is 4/5 of the distance from Point A to Point B, you can use the formula:
x = x1 + (4/5) * (x2 - x1)
y = y1 + (4/5) * (y2 - y1)
z = z1 + (4/5) * (z2 - z1)
Substituting the values:
x = 7 + (4/5) * (7 - 7) = 7
y = -10 + (4/5) * (0 - (-10)) = -10 + (4/5) * 10 = -10 + 8 = -2
z = -4 + (4/5) * (10 - (-4)) = -4 + (4/5) * 14 = -4 + 11.20 = 7.20
Therefore, the point that is 4/5 of the way from (7, -10, -4) to (7, 0, 10) is (7, -2, 7.20).