Mathematics OPERATIONS ON A VECTOR: ADDITION OF TWO VECTORS, SUBTRACTION OF A TWO VECTORS, MULTIPLICATION OF VECTORS BY A SCALAR

Addition Of Vectors :

`text(Triangle law of addition : )`

lf two vectors are represented by two consecutive sides of a triangle
then their sum is represented by the third side of the triangle but in
opposite direction. This is known as the triangle law of addition of
vectors.

Thus, if `vec(AB) = vec(a)` `, vec(BC) =vec(B)` ,and `vec(AC) =vec(c)`
then `vec(AB)+vec(BC) =vec(AC)` i.e. `vec(a)+vec(b)=vec(c)`

Converse of triangle law is also true.

`text(Parallelogram Law of Addition : )`
If two vectors are represented by two adjacent sides of a parallelogram, then their sum is represented
by the diagonal of the parallelogram whose initial point is the same as the initial point of the given vectors.
This is known as parallelogram law of addition of vectors.

Thus if `vec(OA) =vec(a)`, `vec(OB)=vec(B) ` and `vec(OC) =vec(c)`

then `vec(OA) +vec(OB) =vec(OC)`

i.e. `vec(a)+vec(b)=vec(c)`

Position Vectors :

Let `O` be fixed point in space, then vector `vec(OP)` ( `P` is any point in space) is called position vector of
point `P` w.r.t. `O`. If `A` and `B` are any two point in space then

`vec(AB)` = p.v. of `B` - p.v. of `A = vec(OB)- vec(OA)`

i.e. `vec(AB) =vec(b) -vec(a)`

Note: Position vector of a point `P(x, y, z)` in terms of its cartesian coordinate is `vec(OP) = x hat(i) + y hat(j) + z hat(k)`.

Multiplication Of a Vector By Scalar :

If `vec(a)` is a vector & `m` is a scalar, then `m` `vec(a)` is a vector parallel to a whose modulus is
`| m |` times that of `vec(a)` This multiplication is called `text(SCALAR MULTIPLICATION.)` If `vec(a)` & `vec(b)` are vectors &
`m, n` are scalars, then :

`m vec(a) = vec(a)m = m vec(a) quad quad quad ` & `quad quad quad m(nvec(a) ) = n(mvec(a)) = (mn)vec(a)`
`(m + n)vec(a) = mvec(a)+nvec(a)quad quad quad ` & ` quad quad quad m(vec(a) +vec(b)) = m vec(a)+mvec(b) `