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.