At the point were a line intersects a plane [with the equation (24x+32y+40z=480) and three points of A(5, 10, 1), B(6, 3, 6), and C(12, 1, 4)], the vector i-, vector j-, and vector k-coefficients of the line equal the corresponding values of x, y, and z for the plane. By substituting these coefficients, for x, y, and z in the equation for the plane, find the directed distance, d, from the known point (7, 11, 3) to the intersection point (x, y, z).
You have two vectors:
a = (7,11,3)
b = (x,y,z)
The vector from a to b can be found by
a + d = b
so, d = b - a
= (x-7,y-11,z-3)
so the distance is |d| in the direction of d.
To find the directed distance, d, from the known point (7, 11, 3) to the intersection point (x, y, z), we can follow these steps:
Step 1: Determine the direction vector of the line.
To determine the direction vector of the line, we can subtract the coordinates of two points on the line. Let's use point A(5, 10, 1) and B(6, 3, 6):
Direction vector = (6, 3, 6) - (5, 10, 1)
= (1, -7, 5)
So, the direction vector of the line is (1, -7, 5).
Step 2: Substitute the vector coefficients into the equation for the plane.
The equation of the plane is given as 24x + 32y + 40z = 480. We need to substitute the vector coefficients into this equation.
Given vector coefficients:
x = 1
y = -7
z = 5
Substituting these values into the plane equation:
24(1) + 32(-7) + 40(5) = 480
24 - 224 + 200 = 480
-200 + 200 = 480 - 24
0 = 456
Since 0 does not equal 456, the given vector coefficients do not satisfy the plane equation. Therefore, the line does not intersect the plane at the given point.
As a result, we cannot determine the directed distance, d, from point (7, 11, 3) to the intersection point.