Name: Vector Problems
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Chapters

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6
Solution of exercise 7
Solution of exercise 8
Solution of exercise 9

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Exercise 1

Calculate the head of the vector $\text{[math]}$ knowing that its components are $\text{[math]}$ and its tail is $\text{[math]}$ .

Exercise 2

Given points $\text{[math]}$ and $\text{[math]}$ , calculate the value of a if the magnitude of the vector $\text{[math]}$ is one.

Exercise 3

Normalize the vectors: $\text{[math]}$ and $\text{[math]}$ .

Exercise 4

Determine the unit vector, $\text{[math]}$ , which is in the same direction as the vector $\text{[math]}$ .

Exercise 5

Calculate the coordinates of D so that the quadrilateral formed by the vertices: $\text{[math]}$ and D; is a parallelogram.

Exercise 6

The vectors $\text{[math]}$ and $\text{[math]}$ form a basis. Express this in basis the vector $\text{[math]}$ .

Exercise 7

Find the value of k so that the angle that forms between $\text{[math]}$ and $\text{[math]}$ is:

1 $\text{[math]}$

2 $\text{[math]}$

3 $\text{[math]}$

Exercise 8

Calculate the value of a so that the vectors $\text{[math]}$ and $\text{[math]}$ form an angle of $\text{[math]}$ .

Exercise 9

If $\text{[math]}$ is an orthonormal basis, calculate:

1 $\text{[math]}$

2 $\text{[math]}$

3 $\text{[math]}$

4 $\text{[math]}$

Solution of exercise 1

Calculate the head of the vector $\text{[math]}$ knowing that its components are $\text{[math]}$ and its tail is $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Solution of exercise 2

Given points $\text{[math]}$ and $\text{[math]}$ , calculate the value of a if the magnitude of the vector $\text{[math]}$ is one.

$\text{[math]}$ $\text{[math]}$

Solution of exercise 3

Normalize the vectors: $\text{[math]}$ and $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Solution of exercise 4

Determine the unit vector, $\text{[math]}$ , which is in the same direction as the vector $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$

Solution of exercise 5

Calculate the coordinates of D so that the quadrilateral formed by the vertices: $\text{[math]}$ and D; is a parallelogram.

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Hence, $\text{[math]}$

Solution of exercise 6

The vectors $\text{[math]}$ and $\text{[math]}$ form a basis. Express this in basis the vector $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Replacing the value of a in the second equation:

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Plugging the value of b in the first equation:

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

$\text{[math]}$

Solution of exercise 7

Find the value of k so that the angle that forms between $\text{[math]}$ and $\text{[math]}$ is:

1 $\text{[math]}$

$\text{[math]}$ $\text{[math]}$

2 $\text{[math]}$

$\text{[math]}$ $\text{[math]}$

3 $\text{[math]}$

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

After solving the above equation:

$\text{[math]}$

Solution of exercise 8

Calculate the value of a so that the vectors $\text{[math]}$ and $\text{[math]}$ form an angle of $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Solution of exercise 9

If $\text{[math]}$ is an orthonormal basis, calculate:

1 $\text{[math]}$

2 $\text{[math]}$

3 $\text{[math]}$

4 $\text{[math]}$

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Vector Problems

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Solution of exercise 7

Solution of exercise 8

Solution of exercise 9

Theory

Linearly Independent Vectors

Unit Vector

Angle Between Two Vectors

Vector Projection

Scalar Multiplication

Vector Addition

Cross Product

Dot Product

Adding Vectors

Direction Cosine

Linearly Dependent Vectors

Basis Vector

3D Vectors

Linear Dependence and Independence

Scalar Triple Product

Standard Basis

Orthogonal and Orthonormal Vectors

Vectors in the Plane

Linear Combination of Vectors

Cartesian and Polar Coordinates

Scalar Product of Vectors

Formulas

Vector Formulas

Exercises

Cross Product Problems

Dot and Triple Product Problems

Vector Problems

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