# Vector Problems

### Solutions

1Calculate the head of the vector knowing that its components are (3, −1) and its tail is A = (−2, 4).

2Given points A = (0, a) and B = (1, 2), calculate the value of **a** if the magnitude of the vector is one.

3Normalize the vectors: = (1, ), = (−4, 3) and = (8. −8).

4Determine the unit vector, , which is in the same direction of the vector .

5Calculate the coordinates of D so that the quadrilateral formed by the vertices: A = (−1, −2), B = (4, −1), C = (5, 2) and D; is a parallelogram.

6The vectors = (1, 4) and = (1, 3) form a basis. Express this in basis the vector = (−1. −1).

7Find the value of **k** so that the angle that forms between = (3, k) and = (2, −1) is:

1 90°

2 0°

3 45°

8Calculate the value of **a** so that the vectors = 3 + 4 and = a − 2 form an angle of 45°.

9 If { , } is an orthonormal basis, calculate:

1 ·

2 ·

3 ·

4 ·

## 1

Calculate the head of the vector knowing that its components are (3, −1) and its tail is A = (−2, 4).

3 = x_{B} − (−2)x_{B} = 1

−1 = y_{B} − 4y_{B} = 3

B(1, 3)

## 2

Given points A = (0, a) and B = (1, 2), calculate the value of **a** if the magnitude of the vector is one.

## 5

Calculate the coordinates of D so that the quadrilateral formed by the vertices: A = (−1, −2), B = (4, −1), C = (5, 2) and D; is a parallelogram.

## 6

The vectors = (1, 4) and = (1, 3) form a basis. Express in this basis the vector = (−1. −1).

(−1. −1) = a (1, 4) + b (1, 3)

−1 = a +b a = −1 −b a= 2

−1 = 4a +3b −1 = 4( −1 −b) +3b b = −3

= 2 − 3

## 7

Find the value of **k** so that the angle that forms between = (3, k) and = (2, −1) is:

1 **90°**

2 **0°**

3 **45°**

## 8

Calculate the value of **a** so that the vectors = 3 + 4 and = a − 2 form an angle of 45°.

## 9

If { , } is an orthonormal basis, calculate:

1 · = 1 · 1 · cos 0° = 1

2 · = 1 · 1 · cos 90° = 0

3 · = 1 · 1 · cos 90° = 0

4 · = 1 · 1 · cos 0° = 1