2.10 Vector equations

Example. x − (x × a) × b = c

. Strategy: take the dot or cross of the equation

with suitable vectors. The equation can be expanded to form

x − (x · b)a + (a · b)x = c.

Dot this with b to obtain

x · b − (x · b)(a · b) + (a · b)(x · b) = c · b

x · b = c · b.

Substituting this into the original equation, we have

x(1 + a · b) = c + (c · b)a

of solutions.