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3. Consider the points A = (-7,2,5), B = (-4, 1, 3) R3. (a) How many planes pass through the two points A, B? Find three different planes along with their general equations that pass through both of these points, or explain why this is impossible. (b) How many lines pass through the two points A, B? Find three different lines along with corresponding vector equations that pass through both of these points, or explain why this is impossible. (c) Show that every point on each of the lines you found in b) ies on each of the planes you found in (a). (d) Based on your work in (a)-(c), complete the following statement: If two distinct points are on a plane, then any line passing through both of those points is the plane.\"

Solution

(a) You require 3 points to define a plane. Here, only 2 points A(-7,2,5) and B(-4,1,3) are given. So, taking the third point as C(u,v,w)), Equation of plane passing throgh A,B and C is given by:

DETERMINANT: x+7   y-2    z-5

                        -4+7 1-2     3-5      = 0.

                         u+7 v-2      w-5

We can, in fact, evaluate this determinant and equate to 0.

This will be a linear equation in x,y,z. Since u,v,w can taken any value, we get infinity of solutions.

So, the answer to the first question is: infinite number of planes pass through the two points given.

(b) The equation of the line passing through A(-7,2,5) and B(-4,1,3) is given by:

(x+7)/(-4+7)   = (y-2)/(1-2)   = (z-5)/(3-5).

That is,

(x+7)/3   = (y-2)/(-1)   = (z-5)/(-2).

It is seen that this defines a unique line.

So, the answer to the second question is: only one line passes through the 2 points given.

(c) On substitution of x,y,z values from the equation of the line on to the equation of the plane given earlier, it is seen they are satisfied. This shows the line in total lies on the plane given above.

(d) It is seen by the results of (c), If distinct points are on a plane, then any line passing through both of these points is on the plane.