Linear Algebra and Its Applications · Problem 3.3.21

Orthogonality — Gilbert Strang

Mathematics · closest-points-on-lines

Problem

Suppose

L_1

is the line through the origin in the direction of

a_1

and

L_2

is the line through

b

in the direction of

a_2

. To find the closest points

x_1 a_1

and

b + x_2 a_2

on the two lines, write the two equations for the

x_1

and

x_2

that minimize

\|x_1 a_1 - x_2 a_2 - b\|

. Solve for

x

a_1 = (1, 1, 0)

a_2 = (0, 1, 0)

b = (2, 1, 4)