A vector can also be defined as an element of a vector space. Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a two-vector, an n-dimensional vector is often called an n-vector, and so on.
The important formulas of vectors are given below:
1. The position vector of any point p(x,y) is
or OP = ( x,y ).
2.The magnitude of position vector
3. The unit vector = where the magnitude of unit vector is 1
Or,the unit vector =
4.The two vectors and are parallel if and where k and m are the scalars.
5.If then is the result vector which is the triangle law of vector addition.
6. The scalar or dot product of any two vectors .
7. The angle between two vectors is
8. and , then :
9. If the position vector of A is , position vector of point B is and position vector of mid-point M is m then
10. If the point P divides Ab internally in the ratio m:n then position vector of P is given by which is a section formula.
11.If P divides AB externally in the ratio m:n then
PRODUCT OF TWO VECTORS
1.Scalar Product ( dot product )
Let then dot product of & is devoted by read as dot and defined by
The scalar product of & is devoted by ,
where being angle between &
& are perpendicular if =
2.Properties of Scalar Product
v. If then
3.Vector (cross) Product of two vectors.
Let be two vectors then the cross product of is devoted by and defined by
We can define in terms of determinants as follows
Note:1 being angle between
Note:3 If , the and & are parallel if .
4. Properties of cross product
v. is perpendicular to both and
vi. is a Area of paralelogram with sides and
vii. = area of triangle having , , as position vectors of vertices of a triangle.
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