## Linear Algebra and Its Applications, Exercise 3.1.3

Exercise 3.1.3. In the $x$ $y$ 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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