**Problem1 :**Find a vector which is perpendicular to both

*u*= (3, 0, 2) and

*v*= (1, 1, 1) .

*u*×

*v*is perpendicular to both

*u*and

*v*, so we need only compute the cross product to do this problem. From the component formula,

*u*×

*v*= (- 2, - 1, 3). Using the dot product we can check that this vector is indeed perpendicular to

*u*and

*v*.

**Problem2 :**A triangle has two sides of length 5 and 6. If the triangle's area is 12, what is the angle between these two sides?

*u*and

*v*with magnitudes 5 and 6, respectively. From the geometric formula for the cross product, we know that the area of the parallelogram defined by these vectors is given by

*A*= |

*u*||

*v*| sin

*θ*= 30 sin

*θ*. The area

*A*of the parallelogram is exactly twice the area of the triangle in question. Hence we can solve for sin

*θ*= 24/30 = 4/5 .