Suppose v1; v2; v3 2 R4. The function

f : R4 ! R : x 7! det

v1 v2 v3 x

is linear. (Recall, the determinant is linear in one row if the other rows are xed. This is

true for the columns, too, because detA = detAT .) Since f is linear, we can represent it

with a matrix and write f(x) = aTx. The vector a, denoted a = v1 v2 v3, is called the

cross product.

a. Explain why a = v1 v2 v3 is orthogonal to v1, v2, and v3.

b. Interpret the function

g : R4 ! R : x 7!

(v1 v2 v3)Tx

geometrically.

c. Find

a =

2

664

1

2

0

-1

3

775

2

664

4

2

1

4

3

775

2

664

0

0

-1

2

3

775

:

1

Suppose v1,v2, V3 E R4. The functibn f:R4>R:xr>det [v1 v2 v3 74] is linear. (Recall, the determinant is linear in one row if the other rows are xed. This is true for the columns, too, because detA : det AT.) Since f is linear, we can represent it with a matrix and write f [x] : aTx. The vector 0, denoted a : w x v; X v3, is called the cross product. a. Explain why a : 1)] X v2 >]R:XI> [v1xvzxv3JTx geometrically. c. Find 1 4 0 (1 _ 2 X 2 X 0 _ 0 1 1 1 4 2