Vectors

Vector
A vector has a magnitude and direction. Vectors can also be called as directed numbers.

Triangle Law
If two vectors are taken in order, then their resultant would be the third side of triangle formed by joining the tail of first and head of second.

Parallelogram Law
If two vectors have a common source of origin or intersection, then their resultant is given by the diagonal of the parallelogram formed by extending parallel sides.

Unit Vector
A vector having magnitude one (unity) is called a Unit Vector. So Unit vector is used to specify the direction of another vector.

Scalar Product
Scalar Product is commutative

a.b = abcosθ

Vector Product
Vector Product is not commutative.

axb = abcosθ

Area
Area of Triangle = 1/2 AB x AC

Area of parallelogram = 1/2 axb

Collinearity of Vectors
If a vector can be expressed as a scalar multiple of the other, then the vectors are collinear

Linear Combination of Vectors
If two vectors are co-planar, then they can be expressed int he linear form , e.g. r = px + qy.

Any two vectors are always co-planar. for three co planar vectors, we need their linear combination.

Section Formula
Internal Division : Position Vector = (mb + na) / (m+n)

External Division : Position Vector = (mb - na) / (m-n)

Mid Point = ( a + b ) / 2

Centroid Formula
Position Vector = (a+b+c)/3

Scalar Triple Product
if a, b , c are three vectors , then

[a b c] = a.(bxc) =

a1a2  a3

b1 b2b3

c1 c2   c3

(Multiplied by [a b c] if non-coplanar vectors)

If any 1 is zero vector,              [a  b c ] = 0

If any two are collinear or equal, [a b c ] = 0

The vectors are coplanar if, [a b c] = 0

If all three are interchanged, the scalar triple product remains same.

If any two are interchanged, the sign changes.

Parallelopiped & Tetrahedron
Volume of Parallelopiped = [a b c]

Volume of Tetrahedron = A - BCD = 1/6 [AB AC AD]

A,B,C,D are coplanar if [AB AC AD] = 0

Vector Geometry
The sum of all vectors taken in order of a closed figure, is Zero.