Linear Algebra

5 questions cover the following:

Determine the inner product of two vectors, the length of a vector in an inner product space, and the cosine of the angle between two vectors.
Determine the cross product of two vectors in 3-space

1. Find the cosine of the angle between the following pairs of vectors and state whether the angle is acute, obtuse or 90 degrees.

a. u=(3,3,3),v=(1,0,4)

b. u=(6,1,4),v=(2,0,-3)

c. u=(2,4,-8),v=(5,3,7)

2. Find the area of the parallelogram determined by the vectors u and v.

a. u=(1,-1,2),v=(0,3,1)

b. u=(2,3,0),v=(-1,2,-2)

3. Find the area of the triangle in 3-space that has the given vertices. P1(2, 6, -1), P2(1, 1, 1), P3(4, 6, 2)

4. Use the cross product to find the sine of the angle between the vectors.

u = (2, 3, -6) and v = (2, 3, 6)

5. Let v = (1, 0, 2, 1, 3). Find the norm of v, a unit vector that has the same direction as v, and a unit vector that is oppositely directed to v.