(i) Zero or Null Vector :A vector whose initial and terminal points are coincident is called zero or null vector
(ii) Unit Vector :A vector whose magnitude is unity is called a unit vector which is denoted by n^
(iii) Free Vectors: If the initial point of a vector is not specified, then it is said to be a free vector.
(iv) Negative of a Vector A vector having the same magnitude as that of a given vector a and the direction opposite to that of a is called the negative of a and it is denoted by —a.
(v) Like and Unlike Vectors Vectors are said to be like when they have the same direction and unlike when they have opposite direction.
(vi) Collinear or Parallel Vectors Vectors having the same or parallel supports are called collinear vectors.
(vii) Coinitial Vectors Vectors having same initial point are called coinitial vectors.
(viii) Coplanar Vectors A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Otherwise they are called non-coplanar vectors
(iX) Equal vectors: Two vectors a and b are said to be equal written as a = b, if they have same magnitude and same direction regardless of the there initial point
Let a and b be any two vectors. From the terminal point of a, vector b is drawn. Then, the vector from the initial point of a to the terminal point B of b is called the sum of vectors a and b and is denoted by a + b. This is called the triangle law of addition of vectors
c= a + b
Properties of Vector addition
(1) Vector addition is commutative i.e. A+B=B+A
(2) Vector addition is associative i.e. A+B+C=(A+B)+C=A+(B+C)=(A+C)+B
(3) A +0= A
4) A+ (-A)=0
Let a and b be any two vectors.
Now a-b = a+ (-b)
So we will first reverse the direction of vector b and then follow the vector addition process
From the terminal point of a, vector -b is drawn. Then, the vector from the initial point of a to the terminal point B of -b is called the sum of vectors a and -b and is denoted by a - b.
Another method to find substraction of vectors would be
Let draw vector a and vector b from the same initial point. And then draw the line from end point of vector b to vector a.This will give a-b
(i) |k a| = |k| |a|
(ii) k O = O
(iii) m (-a) = – ma = – (m a)
(iv) (-m) (-a) = m a
(v) m (n a) = mn a = n(m a)
(vi) (m + n)a = m a+ n a
(vii) m (a+b) = m a + m b
We can represent any vector in rectangular components form. Let us assume an xyz coordinate plane and unit vector i,j and k are defined across x,y,z respectively
Then we can represent any vector in the components forms like
r= xi+yj+ zk
Important take aways
li+mj+nk=(cosθx) i +(cosθy) j+(cosθz) k is the unit vector in the direction of the vector and θx,θy,θz are the angles which the vector makes with x,y and z axis
Addition,substraction and multiplication,equality in component form can be expressed
In component form addition of two vectors is
C = (Ax+ Bx)i + (Ay+ By)j + (Az+ Bz)k
Where, A = (Ax, Ay, Az) and B = (Bx, By, Bz)
Thus in component form resultant vector C becomes,
Cx = Ax+ Bx : Cy = Ay+ By : Cz = Az+ Bz
In component form substraction of two vectors is
D = (Ax- Bx)i + (Ay- By)j + (Az- Bz)k
where, A = (Ax, Ay, Az) and B = (Bx, By, Bz)
Thus in component form resultant vector D becomes,
Dx = Ax - Bx : Dy = Ay- By : Dz = Az- Bz
Equality of vector in components form
Axi +Ayj+Azk= Bxi+Byj+Bzk
Multiplication of scalar by vector in components form
=(kBx)i + (kBy)j+ (kBz)k
In the coordinate system, the line joining the origin O to the point P in the system is called the position vector of Point P
Vector joining two points in the Coordinate system
Let Point P and Q are there
Postion Vector of Point P
OP= x1 i+y1 j+z1k
Postion Vector of Point Q
OQ= x2 i+y2 j+z2k
Then Vector PQ is defined as
=(x2 -x1) i +(y2 -y1) j +(z2 -z1) k
Vectors a and b are collinear, if a = λb, for some non-zero scalar λ.
Let A, B, C be any three points.
Points A, B, C are collinear <=> AB, BC are collinear vectors.
<=> AB = kBC for some non-zero scalar k
Let A and B be two points with position vectors a and b, respectively
(i) Let P be a point dividing AB internally in the ratio m : n. Then,
r = m b + n a / m + n
where r is the position vector of point P
(ii) The position vector of the mid-point of a and b is a + b / 2.
(iii) Let P be a point dividing AB externally in the ratio m : n. Then,
r = m b + n a / m + n
Multiplication of Vectors can be done in two ways
A.B = |A| |B| cosθ
where θ is the angle between two vectors and 0 =< θ <= 180
A.B = (Axi + Ayj + Azk) . (Bxi + Byj + Bzk) = AxBx + AyBy + AzBz.
Thus for calculating the dot product of two vectors, first multiply like components, and then add.
If i , j and k are mutually perpendicular unit vectors i , j and k, then
i * i = j * j = k * k =1
and i * j = j * k = k * i = 0
Angle between Two Vectors If θ is angle between two non-zero vectors, a, b, then we have
a * b = |a| |b| cos θ
cos θ = a * b / |a| |b|
A x B = |A| |B| sinθ n ?
where n ? is the unit vector pointing in the direction perpendicular to the plane of both A and B. Result of vector product is also a vector quantity.
A x B = (Axi + Ayj + Azk) x (Bxi + Byj + Bzk)
= (AyBz - AzBy)i + (AzBx - AxBz)j + ( AxBy - AyBx)k.
Vector product of two vectors can be made to undergo dot or cross product with any third vector.
(a) Scalar tripple product:-
A . (B x C) = B . (C x A) = C . (A x B) obtained in cyclic permutation.
(b) Vector Triple Product:-
A x (B x C) = B(A . C) - C(A - B)