# what is the magnitude of the difference vector?

I read this question in Sears and Zemansky's University Physics book:

"Two displacement vectors, S and T, have magnitudes S = 3 m and T = 4 m. Which of the following could be the magnitude of the difference vector S - T ?(There may be more than one answer.) i. 9 m; ii. 7 m; iii. 5 m; iv. 1 m; v. 0m; vi. -1m "

My question is: shouldn't the direction of both vectors be specified in order for me to solve it? Or do I just assume that S and T are positive and negative respectively as per the equation S - T

• The question is asking you what the possible values of $\mathbf S - \mathbf T$ could be for all possible directions. So for example if one of the options was $666$m you need consider if there is any possible arrangement of the two vectors that could give $|\mathbf S - \mathbf T| = 666$. This freedom to orient the vectors in any direction means more than one of the answers can be correct. – John Rennie Jan 12 '17 at 16:20
• That's what I thought initially, but I had doubts. Thanks for confirming! – DigiNin Gravy Jan 12 '17 at 16:23
• Re Answer (vi): can the magnitude of a vector be negative? – DJohnM Jan 12 '17 at 17:18
• @DJohnM I know that the magnitude can never be negative, but the vector can. – DigiNin Gravy Jan 12 '17 at 18:20
• @DigiNinGravy A vector which is anti-parallel to, say, the $x$-axis of a coordinate system is sometimes called "a vector in the negative-$x$ direction." But it is an incorrect oversimplification to call such an object "a negative vector." – rob Jan 13 '17 at 7:26

The magnitude of the difference vectors depends on the orientation of $$\bf\vec{S}$$ and $$\bf \vec{T}$$. If they are parallel then $$|\bf \vec{S}-\bf \vec{T}|=|\,|\bf \vec{S}|-|\bf \vec{T}|\,|$$ and if they are anti-parallel then $$|\bf \vec{S}-\bf \vec{T}|=|\bf \vec{S}|+|\bf \vec{T}|$$.

Thus the possible values of $$|\bf \vec{S}-\bf \vec{T}\|$$ lie in the range:

$$|\,|\bf \vec{S}|-|\bf \vec{T}|\,| \leq |\bf \vec{S}-\bf \vec{T}| \leq |\bf \vec{S}|+|\bf \vec{T}|$$

I'll let you work out which answers comply with this.

• |S - T| = 5 or | |S| - |T| | = -1 or |S| + |T| = 7. I just don't understand how |S - T| = |S| + |T|. I mean to have |S| + |T| both vectors must be parallel - having the same direction - so | |S| - |T| | seems off, unless we consider T as -T? – DigiNin Gravy Jan 12 '17 at 17:09
• @DigiNinGravy I think you gave the aswer yourself both $\bf T and$-\bf T$have se same magnitude, and the magnitude does not give the direction of the vector. – Mikael Fremling Jan 12 '17 at 18:10 In above Figure move the end point$\:\mathrm{B}\:$of the vector$\:\mathbf{S}\:$around a circle of radius$\:|\mathbf{S }|=3\:$. Try to find the length$\:|\mathbf{S}-\mathbf{T}|\:$of the vector$\:\mathbf{S}-\mathbf{T}\:$when$\:\mathrm{B}\:$is on points$\:\mathrm{P_{ii}},\mathrm{P_{iii}},\mathrm{P_{iv}}\:\$.

• Thanks, but you reversed the magnitudes of S and T. Can you tell me the name of the program you used? – DigiNin Gravy Jan 12 '17 at 18:17
• @DigiNin Gravy : GeoGebra – Frobenius Jan 12 '17 at 18:35