10.21

Consider vectors:
 A = 2i - 3j + ak
 B = bi + j - 4k
 C = 3i + cj +2k

 Vector A is perpendicular to B as well as to C. It is also known that B * C = 2. Use any method studied in this chapter to solve for the three unknowns, a, b, and c.


Matlab code:

% A * B = 0 ... 2bi -3j - 4ak = 0 ... 3j

% A * C = 0 ... 6i - 3cj + 2ak = 0... 6i

% B * C = 2 ... 3bi + cj - 8k = 2k...  10k

%       a      b      c

A = [-4 2 0; 2 0 -3; 0 3 1]

b = [3; -6; 10]

x = A\b

display(A*x)

First take the dot products. Since A is perpendicular to B as well as to C we can say that they are orthogonal so their dot products are 0.  We then pull the vectors 3j, 6i, and 8k from the dot products, leaving us with a b matrix of [3;6;10]. then solves for the unknowns, giving us A=0.5250, B=2.5500, and C=2.3500.  We confirm the result by multiplying A and x, reproducing b

A =

    -4     2     0

     2     0    -3

     0     3     1

b =

     3

    -6

    10

x =

    0.5250

    2.5500

    2.3500

ans =

    3.0000

   -6.0000

   10.0000