# Cross Product Problems

### Solutions

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.