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 Cross Product Exercise and Cross Product Exercise.

3Given the vectors Cross Product Problems and Vector V, find the product Vector Product and verify that this vector is orthogonal to Vector U and Vector V. Also, find the vector Vector Product and compare it with Vector Product.

4Consider the following figure:

Parallelogram

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.

Cross Product Calculations

Cross Product Calculations

Cross Product Operations

Cross Product Solution


2

Find a unit vector that is perpendicular to Cross Product Exercise and Cross Product Exercise.

Cross Product Calculations

Cross Product Operations

Cross Product Solution


3

Given the vectors Cross Product Problem and Cross Product Problem, find the product Cross Product Problem and verify that this vector is orthogonal to Vector U and Vector V. Also, find the vector Vector Product and compare it with Vector Product.

Cross Product Calculations

Orthogonal Vectors

Scalar Product

Perpendiculars

Scalar Product

Cross Product Calculations

Cross Product Solution


4

Consider the following figure:

Parallelogram

Determine:

1 The coordinates of D if ABCD is a parallelogram.

Cross Product Calculations

Cross Product Calculations

Cross Product Operations

Cross Product Calculations

Cross Product Solution

2 The area of the parallelogram.

Area of the Parallelogram

Cross Product Calculations

Cross Product Operations

Cross Product Solution


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 Vector AB and Vector AC are linearly dependent and have proportional components.

Vector AB

Cross Product Calculations

Cross Product Solution

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:

Cross Product Calculations

Cross Product Calculations

Cross Product Calculations

Cross Product Operations

Cross Product Solution


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.

Triangle

Angle Cosine Calculations

Angle Cosine Calculations

Angle Cosine Calculations

Vector Cosines

Vector Cosines

Vector Cosine Solution

2. Calculate the area of the triangle.

Triangle Area

Cross Product Operations

Cross Product Solution