,
Vectors : General Properties
 

[notation][unit vector][ 2D][ 3D ][ addition ][ scalar mult.]

 

 

 

 

Notation

A non-zero vector is has the magnitude of a positive real number and a direction in space.

A vector may be represented by two letters describing a line.The order of the letters indicates the direction and the length of the line its magnitude.

vector notation

An alternative to this notation is to use a single bold letter, for example C. Then the magnitude is |C| or C.

 

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The Unit Vector

A unit vector eg a , has a magnitude of one |a|=1 and can point in any direction.

Sometimes a unit vector is written with an accent over it â.

Different unit vectors point in different directions.

Hence, if F is an ordinary vector,

unit vector notation

a unit vector in the direction of F (^ circumflex accent)

 

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2D representation

P is a point in the x-y plane with coodinates (x,y). i is the unit vector along the x-axis and j is the unit vector along the y-axis.

With Q at (x,0) and R at (0, y):

vector components

 

2D vector representation

 

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3D representation

P is a point in x-y-z space with coodinates (x, y, z). i is the unit vector along the x-axis, j is the unit vector along the y-axis and z is the unit vector along the z-axis.

vector 3D components

3D representation

 

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Addition(Sum) of vectors

This is also called the Parallelogram or Triangle Law.

If two vectors(a & b) are represented in magnitude and direction by the adjacent sides of a parallelogram from a point, then their resultant(r) is represented in magnitude and direction by the diagonal of the parallelogram passing through the point.

parallelogram law for vectors

parallelogram law equation

If two vectors are represented in magnitude and direction by the adjacent sides of a triangle, taken in order, then their resultant is represented in magnitude but opposite in direction by the third side.

triangle law for vector addition

triangle law equation

 

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Scalar Multiplication

Multiplying a vector by a scalar quantity changes its magnitude but not its direction.

scalar multiplication

 

 

 

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