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Section 12.4: Linear Functions
What is a Linear Function of Two Variables?
In two-variable calculus, a linear function is one whose graph is a plane.
Why is the graph of a linear function a plane?
The defining characteristic of a linear function of one variable was that they had a constant slope. If we are standing on a plane, the slope depends on the direction we walk. But in any given direction, the slope will remain constant as we walk across the plane. Most notably, this is true if we walk in the direction of the postive x-axis or positive y-axis. Therefore, on a plane, the slope ratios โz/โx (with y fixed) and โz/โy (with x fixed) remain constant. Thus we arrive at the following definition for the functions defining a plane:
If a plane has slope m in the x-direction, has slope n in the y-direction, and passes through the point (x 0 , y 0 , z 0 ), then its equation is z = z 0 + m(x โ x 0 ) + n(y โ y 0 ). The plane is the graph of the linear function f (x, y) = z 0 + m(x โ x 0 ) + n(y โ y 0 ). If we write c = z 0 โ mx 0 โ ny 0 , then we can write f (x, y) in the equivalent form f (x, y) = mx + ny + c.
Any plane in R^3 can be uniquely determined by a three points, so long as they do not lie on the same line.
Examples:
- Find the linear function whose graph is the plane through the points (4, 0 , 0), (0, 3 , 0), and (0, 0 , 2).
From a Numerical Point of View
Consider the linear function f (x, y) represented by the following table:
x\y 4 6 8 10 12 5 3 6 9 12 15 10 7 10 13 16 19 15 11 14 17 20 23 20 15 18 21 24 27 25 19 22 25 28 31
One will notice that the values of z given by the above table increase linearly in both the x and y directions. The value of z jumps by 4 units for each 5 units traveled in the direction of increasing x with y fixed, and jumps by 3 units for each 2 units traveled in the direction of increasing y with x fixed. Therefore, we can conclude that the slope in the x-direction is m = โx/โy (y fixed) = 4/5, and that the slope in the y-direction is โz/โy (x fixed) = 3/2.
Examples:
- Find an equation for the linear function with the given values.
x\y 10 20 30 40 100 3 6 9 12 200 2 5 8 11 300 1 4 7 10 400 0 3 6 9
Normal Vectors and the Equation of a Plane
It is also possible to utilize any vector perpendicular to a plane, which we will refer to as a normal vector, and a point P 0 on the plane to write down an equation for the plane.
Let ~n = a~i + b~j + c~k be a normal vector to the plane, let P 0 = (x 0 , y 0 , z 0 ) be a fixed point on the plane, and let P = (x, y, z) be any other point on the plane. Use the figure below to come up with an equation for the plane.
An equation of the plane with normal vector ~n = a~i + b~j + c~k and containing the point P 0 = (x 0 , y 0 , z 0 ) is a(x โ x 0 ) + b(y โ y 0 ) + c(z โ z 0 ) = 0.
Letting d = ax 0 + by 0 + cz 0 (a constant), we can write the equation of the plane in the form
ax + by + cz = d.
Examples:
- Find an equation of a plane that passes through the point (2, โ 1 , 3) and is perpendicular to the vector ~n = 5~i + 4~j โ ~k.
- A plane has equation z = 5x โ 2 y + 7. Find a value of ฮป making the vector ฮป~i +~j + 0. 5 ~k normal to the plane.
