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Addition and Scalar Multiplication of Vectors - Problem 3
So let's wrap up our discussion on vector addition and scalar multiplication, by doing some examples that involve, finding the magnitudes of sums, and differences of two vectors. So whenever you are adding vectors or subtracting vectors, you might be curious whether there is a relationship between to the magnitudes of the sum or difference, and the magnitudes of each of the individual vectors.
Well it's not a simple relationship let me just tell you that right. But first, let's add the two vectors u plus v. Remember you add component-wise so u plus v is going to be 4 plus -1, 3 and -8 plus 4, -4. And the magnitude of 3,-4 is the square root of 3², or 9 plus the square root of -4² or 16. 9 and 16 is 25 and root 25 is 5.
What about u minus v? The magnitude of u minus v, remember you subtract component-wise, so 4 minus -1 gives you 5 and -8 minus 4 gives you -12. So the magnitude of this vector is the square root of 5² or 25, plus -12² or 144. 25 and 144 is 169, the square root of 169 is 13. Now what about the magnitudes of u and v individually?
So for u, I have the square root of 4² is 16, plus -8² or 64, 16 plus 64, that's root 80 which is approximately, I calculated it before, 8.94. And root v is the square root of -1² or 1, plus 4² or 16, so root 1 plus 16 which is root 17 and I calculated that it's approximately 4.12.
Now one thing is for sure that in general, the magnitude of u plus v is not going to equal the magnitude of u plus the magnitude of v, and actually you can see it's way off. If you add these you'll get something a little bit more than 13. In fact you get something like 13.06. So this does not equal the sum of these two, and what abut this, does it equal the difference? The difference is something in the order of a calculated as 4.8-ish, nothing like 13.
So in general don't make the assumption that the absolute value of a sum equals the sum of absolute values, or the absolute value of a difference equals the difference of absolute values, that is not in general true.