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 is its length and is denoted by | |.

If two vectors , are in the same direction then = n. where n is a real number. if 0 < n < 1 then | | < | |
if 1 < n then | | > | |
if n < 0 then || and the direction of is opposite the direction of 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 and that add up to . Vector scalar product Let's have two vectors. Vector scalar product is the formula: other notations for scalar product is  or ( , )
The result from scalar product of two vectors is always a real number.

Scalar product properties

•  =  • n(  ) = (n ) = (n ) where n is number
• ( + ) =  +  If the angle between two verctors , is 90° then  = 0, because cos(90°) = 0  = | |2 because the angle between 2 vectors is 180° and cos(180°) = 1

Vectors Problems

1) If = -1. 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  if | | = 5, | | = 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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