same direction. ** Figure . (a) shows two equal vectors A and B . We can easily check their equality.
Shift B parallel to itself until its tail Q coincides with that of A , i.e. Q coincides with O. Then, since their tips S and P also coincide, the two vectors are said to be equal. In general, equality is indicated * Addition and subtraction of scalars make sense only for quantities with same units.
However, you can multiply and divide scalars of different units. ** In our study, vectors do not have fixed locations. So displacing a vector parallel to itself leaves the vector unchanged. Such vectors are called f ree vectors.
However, in some physical applications, location or line of application of a vector is important. Such vectors are called localised vectors. as A = B . Note that in Fig.
. (b), vectors A ′ and B ′ have the same magnitude but they are not equal because they have different directions. Even if we shift B ′ parallel to itself so that its tail Q ′ coincides with the tail O ′ of A ′ , the tip S ′ of B ′ does not coincide with the tip P ′ of A ′ . .
MULTIPLICATION OF VECTORS BY REAL NUMBERS Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A : λ A = λ A if λ > . For example, if A is multiplied by , the resultant vector A is in the same direction as A and has a magnitude twice of | A | as shown in Fig. . (a).
Multiplying a vector A by a negative number −λ gives another vector whose direction is opposite to the direction of A and whose magnitude is λ times | A |. Multiplying a given vector A by negative numbers, say – and – . , gives vectors as shown in Fig . (b).
The factor λ by which a vector