Vectors

A vector is a mathematical object that has magnitude and direction. With other words it is a line of given length and pointing along a given direction. The magnitude of vector vector a is its length and is denoted by |vector a|.

If two vectors vector a,vector b are in the same direction then vector a = n.vector b where n is a real number.

if 0 < n < 1 then |vector a| < |vector b|
if 1 < n then |vector a| > |vector b|
if n < 0 then vector a || vector b and the direction of vector a is opposite the direction of vector b

Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure.

A vector can be resolved along any two directions in a plane containing it. The figure shows how the parallelogram rule is used to construct vectors vector a and vector b that add up to vector c.

Vector scalar product

Let's have two vectors. Vector scalar product is the formula:

other notations for scalar product is vector avector b or (vector a,vector b)
The result from scalar product of two vectors is always a real number.

Scalar product properties

  • vector avector b = vector bvector a
  • n(vector avector b) = (nvector a)vector b = vector a(nvector b) where n is number
  • vector a(vector b + vector c) = vector avector b + vector avector c

If the angle between two verctors vector a,vector b is 90° then vector avector b = 0, because cos(90°) = 0
vector avector a = |vector a|2 because the angle between 2 vectors vector a is 180° and cos(180°) = 1

Vectors Problems

1) If vector a = -1.vector b what can we say about those two vectors?
Solution: Those two vectors are parallel, with the same magnitude and point to contrary directions.

2) What is the scalar product vector avector b if |vector a| = 5, |vector b| = 7 and the angle between the two vectors is 30°

3) Prove with vectors that for every triangle the lenght of one side is smaller than the sum of the other two sides.


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