all the formulas and sub-formulas in vectors for jee mains
Answers
A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)
If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.
Some key points of vectors:
1) The magnitude of a vector is a scalar quantity
2) Vectors can be multiplied by a scalar. The result is another vector.
3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.
4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.
If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)
2) u + 0 = u
3) u + (-u) = 0 (existence of additive inverses)
4) c (du) = (cd)u
5) (c + d)u = cu + d u
6) c(u + v) = cu + cv
7) 1u = u
8) u + (v + w) = (u + v) + w (the associative law of addition)
Given two vectors u and v, their sum or resultant written as (u+v) is also a vector obtained by first bringing the initial point of v to the terminal point of u and then joining the initial point of u to the terminal point of v giving a consistent direction by completing the triangle OAP. This is termed as the Triangle Law of Addition.
Triangle Law of Addition
Let a and b be any two vectors. From the initial point of a, vector b is drawn and the parallelogram AOCB is completed with OA and OB as adjacent sides. The vector OC is defined as the sum of a and b. This is called the parallelogram law of vectors.
Parallelogram Law of Vectors
For adding more than two vectors, Polygon Law of Addition is used.
Internal and External Division:
1) If A and B are two points with position vectors a and b respectively and C is a point which divides AB internally in the ratio m : n, then the position vector of C is given by OC = (mb + na)/ (m + n). This is termed as internal division.