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.

Answer 1

A vector defines a direction and magnitude of the quantity attached to it. Velocity vector which varies with time gives us acceleration. Two or more forces can rotate a body, but may produce different rotational effects.

Complete answer:
In the given question, we are talking about the vectors, which are directions in space. An object that has both a magnitude and a direction is a vector. Geometrically, as a directed line segment, we can picture a vector whose length is the vector's magnitude and with an arrow indicating the direction. The vector 's direction is from its tail to its head.

We are asked if a vector has any location in space. To answer this question, we will give an explanation which is given as: In general, a vector has no definite place in space because if its magnitude and direction do not change, a vector remains unaffected whenever it is displaced anywhere in space. A position vector, however, has a specific position in space.
Vector of position, a straight line with one end attached to a body and the other end attached to a moving point to define the point's position relative to the body. The position vector will change in length or in direction or in both length and direction as the point travels.

We are asked if a vector can vary with time. To answer this question, we will give an explanation which is given as: As an example, the velocity vector of an accelerating particle varies with time. A vector may vary with time.We are also asked whether two equal vectors $a$ and $b$ at different locations in space necessarily have identical physical effects. To answer this question, we will give an explanation which is given as: Two identical vectors do not necessarily have the same physical effects at different locations in space. For instance, two equal forces acting on a body at two distinct points that can cause a body to rotate about an axis will not produce an equal turning effect.

Note: While answering is this, students should not get confused with the normal vectors and the position vectors. We can also say that the displacement vector which varies with time gives us the velocity of the body. For instance, speed is a scalar quantity which only has a magnitude while velocity is a vector quantity which has both magnitude and direction.