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If components of a vector can be positive OR negative OR zero, why must the magnitude of a vector always be non-negative?

Qmechanic
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Ordinary Owl
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  • you can describe a vector as having a magnitude (positive) and a direction (angle relative to some axis), or as having components along the axes, which can be either positive or negative. –  Jun 13 '16 at 01:37

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A vector is a quantity described by a magnitude and a direction. The magnitude is always +ve or zero. A -ve sign in front of a vector indicates the same magnitude but in the opposite direction. The - sign is part of the direction rather than the magnitude.

Like all vectors, a "resultant vector" is neither +ve or -ve. It has a magnitude (which is >= 0) and a direction.

"Components" are scalars. They are the projections of a vector onto the x and y axes (or other axes). They are not magnitudes, because they can be +ve or -ve (as you note) depending on the angle between the vector and the axes. The + or - sign indicates the direction of the projection along the axis.

"Component vectors" are vectors resolved in the x and y directions which add up to the given vector. Since they are vectors they have magnitude and direction - although the only possible directions are the +x/-x and +y/-y directions. They are not +ve or -ve, because these terms do not apply to vectors.

I apologise that this answer may not be mathematically rigorous. The whole issue is confusing, as your question shows. I am trying to distinguish between the different terms being used.

sammy gerbil
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  • But why do you consider components to be "scalar projections" while you consider resultant magnitude to be a "vector with direction"? Both give distance and an angle on a graph. So shouldn't they both be considered vectors with direction? – Ordinary Owl Jun 13 '16 at 02:14
  • " -ve magnitude indicates the opposite direction" is not really right. The -ve sign is a part of the direction unit vector associated with the vector and not a part of the magnitude. A minus sign before a unit vector indicates that the direction is opposite to the direction of the unit vector. The magnitude remains unchanged when you change the sign. – Yashas Jun 13 '16 at 07:36
  • @OrdinaryOwl : The resultant is a vector. I did not write that the "resultant magnitude is a vector." When you are discussing components you need to be clear about whether you are talking about scalars or vectors. – sammy gerbil Jun 13 '16 at 13:07
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    @YashasSamaga : Thank you for pointing out my error. I have edited my answer. The topic is a tricky one for a physicist to deal with. Perhaps rigorous mathematical definitions are required. – sammy gerbil Jun 13 '16 at 13:14