Labware - MA35 Multivariable Calculus - Single Variable Calculus



Normal Vectors for Linear Function Graphs


A normal vector to a linear function graph is any vector which is perpendicular to that function graph.

One way to find such a vector is to use the fact that two nonzero vectors with a dot product of zero must be perpendicular. If we choose a vector parallel to the linear function graph, we can find a normal vector by finding a vector whose dot product with the first is zero.

For example, for the graph of the function

f(x) = px + k,

we can choose the parallel vector (1, p). For the normal vector N, it must be true that

N(1, p) = 0.

One vector which satisfies this equation and is thus a normal vector is (-p, 1).

We can use the same method for the graph of the implicit function

ax + by = c.

One vector which is parallel to the graph is (b, -a).

(a, b) is a normal vector because its dot product with (b, -a) is 0.



  • 1. Find normal vectors for the graphs of the following explicit functions:
    • f(x) = 0
    • f(x) = 2x
    • f(x) = 2x + 5
    • f(x) = 3x + 4
  • 2. Find normal vectors for the graphs of the following implicit functions:
    • x + y = 1
    • 3x + 4y = 12
    • 4x + 3y = 12
    • 4x + 3y = 1
    • y = 5
    • x = 3
  • 3. How does any general expression for a normal vector to a line depend on k for f(x) = px + k and c for ax + by = c. Why is this?