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Home Vectors Vector Derivatives Theorem 1:
 
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Theorem 1:

The next two theorems show the geometric meaning of the vector derivative.

THEOREM 1

Given a curve X = F(f) in the plane, if F'(t0) ≠ 0 then F'(t0) is a direction vector of the line tangent to the curve at t0.

PROOF

Case 1

The curve is not vertical at t0. The tangent line has slope

10_vectors-223.gif

at t. Therefore the vector

F'(t0) = f1'(t0)i + f2'(t0)j

is a direction vector of the tangent line.

Case 2

The curve is vertical at t0. Then f1'(t0) = 0, so

F'(t0) = f2'(t0)j

is a direction vector of the vertical tangent line. F'(f0) is shown in Figure 10.7.1 for a curve X = F(t).

10_vectors-225.gif

Figure 10.7.1
Home Vectors Vector Derivatives Theorem 1:

Last Update: 2006-11-05