Linear Algebra and Its Applications, Exercise 3.1.3

Exercise 3.1.3. In the xy plane two lines are perpendicular if the product of their slopes is -1. Use this fact to derive the condition for two vectors (x_1, x_2) and (y_1, y_2) being orthogonal.

Answer: The line through the origin and (x_1, x_2) has slope x_2/x_1 and the line through the origin and (y_1, y_2) has slope y_2/y_1. If the two lines are perpendicular we have

(x_2/x_1)(y_2/y_1) = -1

Multiplying both sides of the equation by x_1y_1 we have

x_2y_2 = -x_1y_1

or

x_1y_1 + x_2y_2 = 0

which is the condition for the two vectors (x_1, x_2) and (y_1, y_2) being orthogonal.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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