Cross Product Problems
1Find two unit vectors for (2, −2, 3) and (3, −3, 2) and determine the orthogonal vector for the two.
2Find a unit vector that is perpendicular to 
and 
.
3Given the vectors 
and 
, find the product 
and verify that this vector is orthogonal to 
and 
. Also, find the vector 
and compare it with .
4Consider the following figure:
Determine:
1 The coordinates of D if ABCD is a parallelogram.
2 The area of the parallelogram.
5Given the points A = (1, 0, 1), B = (1, 1, 1) and C = (1, 6, a), determine:
1 What values of a are collinear.
2 Determine if values exist for a so that A, B and C are three vertices of a parallelogram of area 3. If values do exist, determine the coordinates of C:
6A = (−3, 4, 0), B = (3, 6, 3) and C = (−1, 2, 1) are the three vertices of a triangle.
1. Calculate the cosine of each of the three angles in the triangle.
2. Calculate the area of the triangle.
1
Find two unit vectors for (2, −2, 3) and (3, −3, 2) and determine the orthogonal vector for the two.
3
Given the vectors and , find the product and verify that this vector is orthogonal to and . Also, find the vector and compare it with .
4
Consider the following figure:
Determine:
1 The coordinates of D if ABCD is a parallelogram.
2 The area of the parallelogram.
5
Given the points A = (1, 0, 1), B = (1, 1, 1) and C = (1, 6, a), determine:
1 What values of a are collinear.
If A, B and C are collinear, the vectors 
and 
are linearly dependent and have proportional components.
2 Determine if values exist for a so that A, B and C are three vertices of a parallelogram of area 3. If values do exist, determine the coordinates of C:
6
A = (−3, 4, 0), B = (3, 6, 3) and C = (−1, 2, 1) are the three vertices of a triangle.
1. Calculate the cosine of each of the three angles in the triangle.
2. Calculate the area of the triangle.
