## 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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