Mathematics PRODUCT OF TWO VECTORS : VECTOR PRODUCT

Cross Product Of Two Vectors :`

`text(Definition:)`

If `vec(a)` and `vec(b)` be two vectors and `theta (0 le theta le pi )` be the angle between them, then their
vector (or cross) product is defined to be a vector whose magnitude is `ab sin theta` and
whose direction is perpendicular to the plane of `vec(a)` and `vec(b)`
such that `vec(a),vec(b)` and `vec(a) xx vec(b)` form a right handed system.

`quadquadquadquadquad:.` ` vec(a) xx vec(b) =| vec(a) | |vec(b)|sin theta hat(n)`

Where `hat(n)` is a unit vector perpendicular to the plane of
`vec(a)` and `vec(b)` such that `vec(a),vec(b)` and `hat(n)` form a right handed
system.

`text(Properties of Vector Product :)`

`1. ` In general , ` vec(a) xx vec(b) ne vec(b) xx vec(a)`. In fact ` vec(a) xx vec(b) = - vec(b) xx vec(a)`.

`2.` For scalar `m, mvec(a) xx vec(b) =m ( vec(a) xx vec(b)) =vec(a) xx m vec(b)`.

`3.` `vec(a) xx ( vec(b) pm vec(c))= vec(a)xx vec(b) pm vec(a) xxvec(c)`

`4.` If `vec(a) | |vec(b) ` then `theta =0` or `pi => vec(a)xxvec(b)=vec(0)` ( but `vec(a)xxvec(b) =vec(0) => vec(a) =vec(0)` or `vec(b) =vec(0)` or `vec(a) | | vec(b)`) . In particular `vec(a) xxvec(a) =vec(0)`

`5.` If `vec(a) bot vec(b)` then `vec(a)xxvec(b) = |vec(a)| |vec(b)| hat(n)` (or `| vec(a)xxvec(b)| = |vec(a) | |vec(b) |`)

`6.` `hat(i)xxhat(j) = hat(j)xxhat(j) =hat(k)xxhat(k) =0` and `hat(i)xxhat(j)= hat(k), hat(j)xxhat(k)= hat(i)` and

`quadquadquadquadquadquadhat(k)xxhat(i) =hat(j)` (use cyclic system)

`7.` Unit vectorperpendicularto `vec(a)` and `vec(b)` is given by `pm (vec(a)xxvec(b))/( |vec(a) xx vec(b)| )`

`8.` If `theta` is angle between `vec(a)` and `vec(b)` then `sin theta = (vec(a)xxvec(b))/(|vec(a) | |vec(b)|)`

`9.` If `vec(a) = a_1 hat(i) +a_2 hat(j) +a_3 hat(k)` and `vec(b) = b_1hat(i )+b_2 hat(j)+b_3 hat(k)`

`quadquadquadquadquadvec(a)xxvec(b) = |(hat(i), hat(j),hat(k) ),(a_1,a_2,a_3),(b_1, b_2,b_3) | =(a_2b_3 -a_3b_2)hat(i)+ (a_3b_1- a_1b_3)hat(j)+ (a_1b_2 -a_2b_1) hat(k)`

Vector Equation Of a Straight Line :

Vector equation of a straight line passing through a given point `A( vec(a))` and parallel to
a given vector `vec(b)` :

Let `O` be the origin. Let the line pass through a given point `A` whose position vector is `vec(a)`, then `vec(OA) = vec(a)`
Let the given line be parallel to vector `vec(b)`
Let `vec(r)` be the position vector any point `P` on the line, then

`vec(OP) =vec(r)`
Since `vec(AP)` is parallel to `vec(b)` `:.` `vec(AP) =t vec(b)`, where `t` is a scalar.

Now, `vec(OP) =vec(OA) +vec(AP)` `:.` `vec(r) =vec(a) +tvec(b) ` .......(i)

Since for different values of `t`, we get different positions of point `P` on the line, hence (i) is the vector
equation of the required straight line.

Vector equation of straight line passing through two given point `A(vec(a))` and `B(vec(b))`

Let `O` be the origin. Let the line pass through two given point `A` and `B` whose position vectors referred
to `O` be `vec(a)` and `vec(b)` respectively, then

`vec(OA) =vec(a) ` and `vec(OB) =vec(b)`

`:.` `vec(AB)=vec(OB)-vec(OA)=vec(b)-a`

Clearly, the required line passes through `A(vec(a))`

and is parallel to the vector `(vec(b) -vec(a) )`.

Hence the vector equation of the required line is,

`vec(r) =vec(a) +t (vec(b)-vec(a))` or `vec(r)= (1-t)vec(a)+tvec(b)`

Important Note:

(i) Two lines in a plane are either intersecting or parallel conversely two intersecting or parallel lines must be
in the same plane

(ii)However in space we can have two neither parallel nor intersecting lines. Such non coplanar lines are
known as skew lines. If two lines are parallel and have a common point then they are coincident.