Vector Problems

1Calculate the head of the vector 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 Vector is one.

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

4Determine the unit vector, Vector, which is in the same direction of the vector 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 Vector = (1, 4) and Vector = (1, 3) form a basis. Express this in basis the vector Vector = (−1. −1).

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

1 90°

2

3 45°

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

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

1 Vector · Vector

2 Vector · Vector

3 Vector · Vector

4 Vector · Vector


1

Calculate the head of the vector 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 Vector is one.

Vector Calculations

Vector Solution


3

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

Vector Calculations

Vector Calculations

Vector Calculations


4

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

Unit Vector

Unit Vector Solution


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.

QuadrilateralVector Calculations


6


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

Vector = 2Vector − 3Vector


7


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

1 90°

Vector Calculations

Vector Solution

2

Vector Calculations

Vector Solution


3 45°

Vector Calculations

Vector Calculations

Vector Solution



8

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

Vector Calculations

Vector Calculations

Vector Solution



9

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

1 Vector · Vector = 1 · 1 · cos 0° = 1

2 Vector · Vector = 1 · 1 · cos 90° = 0

3 Vector · Vector = 1 · 1 · cos 90° = 0

4 Vector · Vector = 1 · 1 · cos 0° = 1




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