Vector

A vector is a directed line segment. If a vector starts from point A and ends at point B, then the vector is named as , where point A is called as the initial point and point B is named as the terminal point. The length / magnitude of the vector is denoted by .

Two vectors that are parallel and have the same length are said to be equal.

The notation of vectors differs. It can be in one of the forms (but not limited) below:

• Boldface letters: a, b, u, v, F
• Letters with arrowhead: ,
• Underlined letters: a, b, u, v

Position Vector
Let  .
There is one directed line segment, v equals to  whose initial point falls at the origin.
This is the representative of  in standard position . v is called the position vector of  .
Try to move point A or B in the graph below. You will find that the position vector v is changing according to the vector .


Component Form of Vector v
If a two-dimensional vector v is a position vector with its terminal point at ( v₁, v₂ ), then its component form is

v =  < v₁, v₂ >.

If a three-dimensional vector v is a position vector with its terminal point at ( v₁, v₂, v₃ ), then its component form is

v =  < v₁, v₂ , v₃ >.

As such, a zero vector 0 = < 0, 0 > in 2-D and 0 = < 0, 0, 0 > in 3-D.

Length / Magnitude of Vector v 
To find the length / magnitude of vector v, we use the concept of distance, we get

.
Take note that in some books, the notation for the magnitude of vector v is ||v||.

Seeing the formula above, can we deduce that

?
Well, we can only prove it after learning the algebra operations of vectors.


Betty, Created with GeoGebra and Thomas Calculus, Pearson.