Find a linear function whose graph is the plane that intersects the xy-plane along the line y=2x+2 and contains the point (1,2,2).
a little vector analysis makes this relatively easy. Find two points B and C on the line. Let A=(1,2,2)
Form vectors AB and AC. These two vectors lie in the desired plane. Let n be the normal to the plane.
n = AB × AC
n•(v-A) = 0 is the desired plane.
So, pick any two points on the line, say B=(0,2,0) and C=(1,4,0)
AB = -i -2k
AC = 2j -2k
n =
| i j k |
|-1 0 -2 |
| 0 2 -2 |
= 4i -2j + 2k
4i -2j + 2k • (x-1)i + (y-2)j + (z-2)k = 0
2x-y+z = 2
For a fuller explanation, see
www.jtaylor1142001.net/calcjat/Solutions/VPlanes/VPPtLine.htm
To find a linear function whose graph is the plane described, we can use the point-normal form of the equation of a plane. The equation of a plane can be written as:
Ax + By + Cz = D
where A, B, and C are the coefficients, and (x, y, z) are the coordinates of any point on the plane. In this case, we know that the plane intersects the xy-plane along y = 2x + 2. This means that the coefficients A and B can be derived from the slope of the line.
Let's break down the steps to find the linear function:
Step 1: Determine the coefficients A and B:
Since the plane intersects the xy-plane along y = 2x + 2, we can equate the given equation to the general equation of a line y = mx + b. Comparing the two equations, we have m = 2 and b = 2. According to the point-normal form, the coefficients A and B are the coefficients of x and y, respectively. Therefore, A = -m = -2 and B = 1.
Step 2: Determine the coefficient C:
We know that the plane contains the point (1, 2, 2). Substituting these values into the equation Ax + By + Cz = D, we have:
-2(1) + 1(2) + C(2) = D
-2 + 2 + 2C = D
2C = D
To simplify the equation, we can set C = 1, which means D = 2C = 2.
Step 3: Write the final equation:
The equation of the plane (linear function) is:
-2x + y + z = 2
So, the linear function whose graph is the plane that intersects the xy-plane along y = 2x + 2 and contains the point (1, 2, 2) is -2x + y + z = 2.