Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.
G. H. Hardy
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One of the important concepts in mathematics is the concept of zero and here we will consider an example involving zero. One well-known result is if the product of two (or more) quantities is equal to 0, then each of them is equal to zero.
Putting it in mathematical form means, if AB = 0 then either A = 0 or B = 0 or A = 0 = B. For example, if (x − 1)(x − 2) = 0 then we have, either (x − 1) = 0 or (x − 2) = 0, resulting in x = 1 or 2. Point to note, the mathematical word or is used here in the inclusive sense. Similar examples we may consider to explore it further. Is it always true? Is there any exceptions? This is the main discussion point here.
So, is this result always true? As long as we are in the range of secondary level, probably this is true. But if we go beyond that, the result may not hold good. Let’s see how.
To disprove any result in mathematics, one common strategy is to provide counter example. Let’s start with Matrices and then we will consider Vectors. Before we start our actual discussion, let us recall few important definitions. First one in our list is the zero (or null) matrix, denoted by O. If each element of an m × n matrix be 0, the matrix is said to be the zero matrix. Observe the difference between the real number 0 and the zero (or null) matrix O. Few null matrices of various orders are given below:
We now consider three matrices A, B and C as given below:
We can check the following two matrix multiplications (recall that, in case of matrix multiplications, AB means A multiplied by B and that the order is very important, so that AB and BA are different):
From the matrix viewpoint, though AB = O or BC = O but we can see that none of A, B or C is zero! To be specific, none of them is a zero matrix.
Next, vectors are denoted by the bold face letters, like X. A zero (or null) vector (represented by P = 0) means a vector whose magnitude is zero, like |P| = 0. Now consider two vectors A and B and suppose θ is the angle between them. Their dot product is represented by A · B. Note that the dot product of two vectors is a scalar quantity. If their dot product is zero, then can we conclude A = B = 0? Let’s analyse.
Now A · B = 0 means |A||B| cos θ = 0. Three possibilities may come to our mind. They are
- either |A| = 0;
- or |B| = 0;
- or cos θ = 0 (means θ = π/2)
In other words, if A = 0 then A · B = 0; if B = 0 then A · B = 0; even if θ = π/2 (means the two vectors A and B are orthogonal/ perpendicular) then A · B = 0. Hence A · B = 0 not always mean that A = 0 or B = 0.
Next, we will consider their cross product namely A × B, which is a vector quantity. Point of discussion here is, if their cross product is zero can we conclude A = B = 0? Let’s analyse.
With usual notation, A × B = 0 means |A||B| sin θ n = 0, where n is a unit vector perpendicular (or normal) to the plane containing both A and B. Again, three possibilities we need to consider and they are
- either |A| = 0;
- or |B| = 0;
- or sin θ = 0.
In other words, if A = 0 then A × B = 0; if B = 0 then A × B = 0; even if θ = 0 (means the vectors A and B are parallel) then also A × B = 0. Hence A × B = 0 not always mean that A = 0 or B = 0.
What follows from the above discussion is that, whenever we are concluding A = 0 or B = 0 or A = 0 = B from AB = 0, we need to be specific about A and B. Needless to say, if we fail to understand the underlying object we might conclude wrong. Above examples, definitely help us to understand this fact.
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