5. Determine an equation for the plane that is exactly between the points A(-1, 2, 4) and B(3, 1, -4).
To determine an equation for the plane that is exactly between two points, you can use the midpoint formula. The midpoint of two points is the average of their coordinates.
Step 1: Find the midpoint of points A and B.
- The midpoint in the x-coordinate direction is: ( (-1 + 3) / 2 ) = 1
- The midpoint in the y-coordinate direction is: ( (2 + 1) / 2 ) = 1.5
- The midpoint in the z-coordinate direction is: ( (4 + -4) / 2 ) = 0
Therefore, the midpoint of A and B is (1, 1.5, 0).
Step 2: The equation of a plane in three-dimensional space can be represented by the standard form Ax + By + Cz + D = 0, where A, B, C, and D are constants.
We need to find A, B, and C. To do this, we can use the fact that the coefficients of x, y, and z in the plane equation represent the normal vector of the plane.
Step 3: Calculate the direction vector for the line connecting points A and B.
- The direction vector is given by B - A = (3 - (-1), 1 - 2, -4 - 4) = (4, -1, -8).
Step 4: Find the cross product of the direction vector and a vector representing a point on the plane.
- Let's use (1, 1.5, 0) as a point on the plane. The cross product can be found as:
(4, -1, -8) x (1, 1.5, 0) = (-8, 32, 7.5)
Step 5: Normalize the vector found in step 4.
- Divide each component of the vector by the magnitude of the vector to get a unit vector.
Normalized vector = (-8, 32, 7.5) / sqrt((-8)^2 + (32)^2 + (7.5)^2) = (-8/sqrt(1224.25), 32/sqrt(1224.25), 7.5/sqrt(1224.25))
Step 6: Write the equation of the plane.
Using the point-normal form of a plane equation, we have:
-8/sqrt(1224.25)(x - 1) + 32/sqrt(1224.25)(y - 1.5) + 7.5/sqrt(1224.25)(z - 0) = 0
Simplifying the equation gives:
-8(x - 1) + 32(y - 1.5) + 7.5(z - 0) = 0
Therefore, an equation for the plane that is exactly between points A(-1, 2, 4) and B(3, 1, -4) is -8(x - 1) + 32(y - 1.5) + 7.5(z - 0) = 0.