# Dot and Triple Product Problems

### Solutions

1Given the vectors , and , calculate the following:

1. ,

2. ,

3.

4.

5.

2For what values of a do the vectors , and form a basis?

3Determining the value of the coefficient k for the vectors = k − 2 + 3, = − + k + if the vectors are:

1.

2. Parallel.

4Find the direction cosines of the vector .

5Calculate the angle between the vectors and .

6Given the vectors and , calculate:

1 The magnitudes of and ·

2 The cross product of and ·

3 The unit vector orthogonal to and ·

4 The area of the parallelogram whose sides are the vectors and ·

7Calculate the triple product of: if .

8Given the vectors , and , calculate the triple product . Also, what is the volume of the parallelepiped whose edges are formed by these vectors?

## 1

Given the vectors , and , calculate the following:

1. ,

2. ,

3.

4.

5.

## 2

For what values of a do the vectors , and form a basis?

For a ≠ 1, the vectors form a basis.

## 3

Determining the value of the coefficient k for the vectors = k − 2 + 3, = − + k + if the vectors are:

1.

2. Parallel.

The system does not have a solution.

## 4

Find the direction cosines of the vector .

## 5

Calculate the angle between the vectors and .

## 6

Given the vectors and , calculate:

1 The magnitudes of and ·

2 The cross product of and ·

3 The unit vector orthogonal to and ·

4 The area of the parallelogram whose sides are the vectors and ·

## 7

Calculate the triple product of: if .

## 8

Given the vectors , and , calculate the triple product . Also, what is the volume of the parallelepiped whose edges are formed by these vectors?