A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer.

Question

A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer.

Philoid · Accepted Answer

A vector has magnitude and direction but in general it does not have a fixed location of space because a vector can be translated parallel to itself or we can say if a vector is moved parallel to itself keeping its direction and magnitude same then the vector is same or there is no effect on vector as can be shown in figure

Now here there are three vectors at A,B,C all have same length so have same magnitude, have same direction towards positive x axis and are thus parallel to each other so all the three vectors are same so there is no effect of location i.e. position is not fixed but in case of position vector position of each point is different and position vector denotes position in terms of co-ordinates of x and y so position vector have a fixed location and are also directed from origin so two position vectors cannot be parallel if they are denoting different positions so position vector have definite position in space but in general all vectors does not have a specific position in space

Yes , vector can certainly vary with time and many vector quantities are just rate of variation of other quantities or vectors can be a function of time for e.g. Velocity of a particle in uniform motion is a function of time and as the time increases the velocity of particle increases or decreases depending upon the acceleration of particle i.e. velocity changes with time likewise in general vector can vary with time

Now we have two equal vectors a and b at different locations in space they necessarily need not have same physical effects though in specific cases they can have same physical effects but this is not true always for e.g. two forces of same magnitude and same direction applied on a body fixed lever can produce different turning effects as shown in figure

As can be seen both a and b are directed vertically downwards i.e. same direction and have same magnitude so both are equal but turning effect will be different due to them because their location is different in same so we conclude that equal vectors a and b at different locations in space do not necessarily have identical physical effects