Addition of Vectors
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 O 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.
Let a and b be any two vectors. From the initial point of a, vector b is drawn and parallelogram OACB 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 addition of vectors.
The sum of two vectors is also called their resultant and the process of addition as composition.
Properties of Vector Addition
(i) a + b = b + a (commutativity)
(ii) a + (b + c)= (a + b)+ c (associativity)
(iii) a+ O = a (additive identity)
(iv) a + (— a) = 0 (additive inverse)
(v) (k1 + k2) a = k1 a + k2a (multiplication by scalars)
(vi) k(a + b) = k a + k b (multiplication by scalars)
(vii) |a+ b| ≤ |a| + |b| and |a – b| ≥ |a| – |b|
Difference (Subtraction) of Vectors
If a and b be any two vectors, then their difference a – b is defined as a + (- b).
Multiplication of a Vector by a Scalar
Let a be a given vector and λ be a scalar. Then, the product of the vector a by the scalar λ is λ a and is called the multiplication of vector by the scalar.
(i) |λ a| = |λ| |a|
(ii) λ 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