## Working with Vectors

If we divide an arbitrary vector ** a** by its magnitude we obtain a (dimensionless) vector in the same direction as

**and of magnitude 1 (a dimensionless number). Such a vector is known as a unit vector in the direction of vector**

*a***and is denoted by**

*a***=**

*â***/|**

*a***|**

*a*A zero vector is defined so that ** a** + (-1)

**=**

*a*

*0*Addition of vectors is both commutative and associative i.e. ** a** +

**=**

*b***+**

*b***and**

*a***+ (**

*a***+**

*b***) = (**

*c***+**

*a***) +**

*b*

*c*For any scalar *k*, *k a* +

*k*=

**b***k*(

**+**

*a***)**

*b*A vector may be resolved into component vectors along appropriately chosen directions. Given a vector ** a**, its orthogonal component vectors parallel and normal to a direction inclined at an angle

*θ*to

**are of magnitude,**

*a*|*a*_{p}| = |** a**| cos

*θ*and |

*a*_{n}| = |

**| sin**

*a**θ*

The Cartesian unit vectors in the directions of the Cartesian axes *x*, *y*, *z* are denoted by ** i**,

**,**

*j***.**

*k*A vector ** a** can be expressed in Cartesian form as a =

*a*

_{x}

**+**

*i**a*

_{y}

**+**

*j**a*

_{z}

*k*The scalars *a*_{x}, *a*_{y}, *a*_{z} are called the Cartesian scalar components of ** a**, whereas the vectors

*a*

_{x}

**,**

*i**a*

_{y}

**,**

*j**a*

_{z}

**are called the Cartesian component vectors of**

*k***.**

*a*The magnitude of the vector ** a** is given by |

**| = (**

*a**a*

_{x}

^{2}+

*a*

_{y}

^{2}+

*a*

_{z}

^{2})

^{1/2}

The operation of scaling and vector addition take the following Cartesian algebraic forms (for any scalar α and vectors ** a** and

**);**

*b*α*a* = α*a*_{x}** i** + α

*a*

_{y}

**+ α**

*j**a*

_{z}

*k*and

** a** +

**= (**

*b**a*

_{x}+

*b*

_{x})

**+ (**

*i**a*

_{y}+

*b*

_{y})

**+ (**

*j**a*

_{z}+

*b*

_{z})

*k*A vector represented by ** a** =

*a*

_{x}

**+**

*i**a*

_{y}

**+**

*j**a*

_{z}

**can also be represented by the abbreviated notation of a tripled pair**

*k**a*= (

*a*

_{x}

**,**a_{y}

*,**a*

_{z}).

The operation of scaling and vector addition take the following abbreviated forms (for any scalar α and vectors ** a** and

**),**

*b*α*a* = (α*a*_{x}** ,** α

*a*

_{y}

**α**

*,**a*

_{z})

and

** a** +

**= (**

*b**a*

_{x}+

*b*

_{x},

*a*

_{y}+

*b*

_{y},

*a*

_{z}+

*b*

_{z})

Vectors may be used to determine the position of points relative to a chosen origin , and they are then known as position vectors.