Motion in a Plane
Scalars and Vectors
A quantity with magnitude only is called scalar quantity. A quantity with both magnitude and direction is called vector quantity.
Position and Displacement
The following figure shows positions of an object at different times.
Let O be the origin. Let P and P' are positions of the object at times t and t' respectively. Let us join O and P by a straight line. Then OP is the position vector of object at time t. Similarly, OP' is the position vector of object at time t'. Let OP represented by r and OP' represented by r'. If the object moves from P to P' then PP' is displacement vector of object. Displacement vector is the straight line joining the initial and final positions.
The second figure shows different paths of object which are PABCQ, PDQ and PBEFQ. No matter which path the object follows, its displacement vector will always be PQ. So, it can be said that the magnitude of displacement is either less than or equal to the path length of object between two points.
Equality of Vectors
Two vectors A and B are equal if, and only if, they have same magnitude and same direction.
The first figure shows two vectors A and B which are shown by parallel lines OP and QS. If we move OP towards QS, a time will come when O is coincident with Q, and P is coincident with S, it will show that OP and QS can become coincident. This will also show that A is equal to B.
The second figure shows two vectors A' and B' which have equal magnitude but different directions. No matter how much we move one vector towards another, their heads and tails will not become coincident at the same time. So, A' and B' are not equal.
Multiplication of Vectors by Real Numbers
When we multiply a vector A with a positive number λ we get a vector whose magnitude is changed by the factor λ but direction remains the same.
|λA|= λ|A| if λ > 0
When we multiply a given vector by negative number we get another vector with direction opposite to direction of given vector and magnitude is λ times the given vector.
Addition and Subtraction of Vectors:
Vectors obey the triangle law of addition or equivalently the parallelogram law of addition.
Let us assume there are two vectors A and B lying in the same plane. Let us place B so that its tail is at the head of vector A. Now, join the tail of A to head of B by a line OQ which represents the vector R. In this case, R = A + B. The third figure shows R = B + A
Addition of vectors is commutative which means:
A + B = B + A
Addition of vectors obeys the law of associative property, which means:
(A + B) + C = A + (B + C)
When we add two equal and opposite vectors, we get null as result.
A – A = 0
|0| = 0
Direction of null vector cannot be specified. When we multiply a vector by zero, we get null as result.
These figures illustrate the parallelogram law of addition of vectors. Figures b and c show that parallelogram law is equivalent to triangle law of addition. If two vectors A and B are positioned in a way that their tails coincide then we make a parallelogram OQSP; as shown in figure. In this case, the diagonal OS (represented by R) gives the sum of vectors A and B. If head of A is coincident with tail of B then R gives the third side of the triangle OPS. In this case, the triangle law of addition is obeyed.