2Vectors

IA Vectors and Matrices

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

If (1 + a · b) is non-zero, then

x =

c + (c · b)a

1 + a · b

Otherwise, when (1 +

a · b

) = 0, if

c + (c · b)a 6= 0

, then a contradiction is

reached. Otherwise,

x · b = c · b

is the most general solution, which is a plane

of solutions.