: A + B = B + A ( . ) The addition of vectors also obeys the associative law as illustrated in Fig. . (d).
The result of adding vectors A and B first and then adding vector C is the same as the result of adding B and C first and then adding vector A : ( A + B ) + C = A + ( B + C ) ( . ) What is the result of adding two equal and opposite vectors ? Consider two vectors A and – A shown in Fig. .
(b). Their sum is A + (– A ). Since the magnitudes of the two vectors are the same, but the directions are opposite, the resultant vector has zero magnitude and is represented by called a null vector or a zero vector : A – A = | |= ( . ) Since the magnitude of a null vector is zero, its direction cannot be specified.
The null vector also results when we multiply a vector A by the number zero. The main properties of are : A + = A λ = A = ( . ) Fig. .
(a) Two vectors A and B , – B is also shown. (b) Subtracting vector B from vector A – the result is R . For comparison, addition of vectors A and B , i.e. R is also shown.
What is the physical meaning of a zero vector? Consider the position and displacement vectors in a plane as shown in Fig. . (a).
Now suppose that an object which is at P at time t , moves to P ′ and then comes back to P. Then, what is its displacement? Since the initial and final positions coincide, the displacement is a “null vector”. Subtraction of vectors can be defined in terms of addition of vectors.
We define the difference of two vectors A and B as the sum of two vectors A and – B : A – B = A + (– B ) ( . ) It is shown