Vector Problems
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 = xB − (−2)xB = 1
−1 = yB − 4yB = 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
