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I have a line with point a, (6, 12) and point b (45, 18).
To describe this line using vectors, the equation is L = [6, 12] +t [39, 6]
How would I go about finding the normal to this line?
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A 2d line with normal vector \bfn is given by \bfn\cdot \bfx = b for some b. To find \bfn, we should write the given line in this form. Let \bfn = (n_1, n_2) and \bfx = (x, y). The line \bfn \cdot \bfx = b is the same as n_1 x + n_2 y = b.
A unit vector is a vector of length 1. Any nonzero vector can be divided by its length to form a unit vector. Thus for a plane (or a line), a normal vector can be divided by its length to get a unit normal vector. Example: For the equation, x + 2y + 2z = 9, the vector A = (1, 2, 2) is a normal vector.
Formulas for the magnitude of vectors in two and three dimensions in terms of their coordinates are derived in this page. For a two-dimensional vector a=(a1,a2), the formula for its magnitude is ∥a∥=√a21+a22.
From the Math StackExchange How do I calculate the normal vector of a line segment?:
if we define dx=x2-x1 and dy=y2-y1, then the normals are (-dy, dx) and (dy, -dx).
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