Vectors are defined by magnitude and direction. Vectors can be define as, the line segment AB is a vector which is denoted by a and is read as vector a. Point A is called the initial point from where AB starts, point B is called the terminal point where AB ends and the distance between A and B is called as magnitude or length of the vector a and is denoted by ∣a∣. The arrow indicates the direction of the vector.
The position vector of a point P(x,y,z) with respect to origin O (0,0,0) is drawn. Its magnitude will be given by|OP| = x2+y2+z2 and the vector OP will have O and P as its initial and terminal points, respectively. This vector denotes the position vector of the point P with respect to O.
Types of Vectors
Zero Vector: A vector which has got the same initial and ending points. It is denoted by O
Unit Vector: A vector with its magnitude equal to one is called a unit vector. Thus, i^ is a unit vector of I, where ∣i^∣ = 1.
Co-initial vectors: When two or more vectors have the same initial points
Equal Vectors: Two vectors are equal when they have same magnitude and direction. The initial points doesn’t matter here. They may be represented as: b = a
Negative of a Vector: A vector whose magnitude is the same but direction is opposite to the original vector say PQ is called a negative vector of O i.e. QP = – PQ
Free Vectors: The vectors whose initial points are not fixed.
Parallel Vectors: The vectors which may have different magnitude but all should have the same or opposite direction are called parallel vectors.
Collinear Vectors: Vectors which may have same direction or are parallel or anti-parallel. As magnitudes can vary, we can find some scalar vector λ for which a=λb.
Non-collinear Vectors: Two vectors acting in different directions are called non-collinear vectors or independent vectors. Though a and – b have same magnitude, but we can’t express a or – b in terms of one another. Two non-collinear vectors describe a plane.
Co-planar Vectors: Two parallel vectors or non-collinear vectors are always co-planar to one another. Usually, more than two vectors if they lie within the same plane, they are called co-planar vectors.
Important Fundamental Theorems of Vectors
In Two Dimensions:
If there are two non-zero or non-collinear vectors p and q, then any vector x which lies in the plane of p and q can be written as a linear combination of p and q. We can also write this as there exists L and M ϵ R such that L *p + M * q = x.
This also proves that if L1p+M1q=L2p+M2q exists, then we can write
L1=L2 and M1=M2
In Three Dimensions:
If there are three non-zero or non-collinear vectors p,qandr then any vector x which lies in the plane of p,qandr can be written as a linear combination of p,qandr We can also write this as there exists L, M ad N ϵ R such that
This also proves that if L1p+M1q+N1r=L2p+M2q+N2r exists, then we can write L1 = L2, M1 = M2 and N1 = N2
A vector x is said to be a linear combination of vectors a1,a2,a3,…….,an in a way that there exists scalar m1,m2,m3,…….,mn and we can write x=a1m1+a2m2+a3m3+…+anmn
A system of vectors a1,a2,a3,…….,an are said to be linearly independent
If x=a1m1+a2m2+a3m3+…+anmn=0 which means that
A pair of non-collinear vectors is linearly independent.
A triad of non-coplanar vector is linearly independent.
A set of vectors a1,a2,a3,…….,an is said to be linearly dependent such that scalars m1,m2,m3,…….,mn exists such that all of the scalars are not equal to zero and
A pair of collinear vectors is linearly independent.
A triad of coplanar vector is linearly independent.
If aandb be two non-collinear vectors, then every vector i which is co-planar with aandb can be expressed in one and only one combination in the form of xa+yb=i where x and y are scalar components of the respective vectors.
If two vectors are perpendicular to each other, then the vectors can be supposed to be drawn along the X-axis and Y-axis respectively. If the unit vectors along the two vectors is represented by i^ respectively, then we can write that
If a,bandc are non-coplanar vectors, then any vector i can be uniquely expressed as a linear combination i=xa+yb+zc where x, y and z are scalar components of the respective vectors.
If vectors a=a1i^+a2j^+a3k^;
are co-planar, then
Three points with position vectors a,bandc are collinear if and only if there exist scalars like x, y and z all of which is not equal to zero such that
xa+yb+zc=0 and x + y + z = 0
Four points with position vectors a,bcandd are coplanar if and only if there exist scalars like x, y, z and w (sum of any two is not equal to zero) such that
xa+yb+zc+wd=0 and x + y + z + w = 0
Question 1: Points P (p), Q (q), R (r) and S (s) are related as ap+bq+cr+ds=0), and a + b + c + d = 0 (where a, b, c, d are scalars and sum of any two is not zero). Prove that if P, Q, R and S are concyclic then
From the question given, it is understood that P, Q, R and S are coplanar.
Now they are concyclic also.
So, we can write
PO x QO = RO x SO
Question 2: Vectors vecp and vecq are non-collinear. Find for what value of y vector vecw=(x–2)p+q and a=(2x+1)p–q are collinear?