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.

Distance Between Two Points

In xy-plane, the distance between P₁(x₁, y₁) and P₂(x₂, y₂) is calculated using Pythagoras theorem,

.
In space (three-dimensional system), how do we find the distance between point A and point B? Let's view the video below to find it out. If you want to play around with the points, just click here.

From the video, we conclude that the distance between point A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is

.
As such, if P₀(x₀, y₀, z₀)  is a fixed point and P(x, y, z) is a moving point that moves at a constant distance of a units from P₀, we'll get a sphere centered at (x₀, y₀, z₀) with radius a units. The equation of the sphere can be obtained from the concept of distance, i.e.

.
You may view the video below to see the forming of the equation using the distance concept.

You may also drag the point P below to see the locus of it. If you want to rotate the 3-D view to have better learning experience, just click here.

Betty, Created with GeoGebra, Screencast-O-Matic and Thomas Calculus, Pearson.