Suppose a particle undergoes a displacement A followed
by a second displacement B. The
final result is the same as if the particle had started at the same
initial point and undergone a single displacement C. We call displacement C the
vector sum, or resultant, of displacements A and B. We express this relationship
symbolically as
C = A + B
The boldface plus sign emphasize that adding two vector
quantities require a geometrical process and is not same process as adding two
scalar quantities as 2 +3 =5. In vector addition we usually place the tail of
second vector at the head, or tip, of first vector. If we make the displacement A and B in reverse order, with B first
and A second, the
result is same. Thus,
C = B + A and A
+ B = B + A
This shows that the order of terms in a vector sum doesn’t matter.
In other words, vector addition obeys the commutative law.
If A and B are both drawn with their tails at the same
point, vector C is the diagonal of a parallelogram
constructed with A and B as two adjacent sides.
It’s a common error to
conclude that if C = A + B, then the magnitude C should equal the magnitude
A plus the magnitude B. In general, this conclusion is wrong; for the
vectors shown in figure above, you can see that C < A +B. The magnitude of A + B depends on the magnitude of A and B and on the angle between A and B. Only in the special case in which A and B are parallel is the magnitude of A + B equal to the sum of the magnitudes of A and B. When the vectors are anti-parallel, the magnitude
of C equals the difference of the magnitudes of A and B. Be careful about distinguishing
between scalar and vector quantities, and you’ll avoid making errors about
the magnitude of a vector sum. When we need to add more than two
vectors, we may first find the vector sum of any two, add this vectorially
to the third, and so on.
We can subtract vectors
as well as add them. To see how, recall that vector -A has has the same magnitude as A but the opposite direction. We define the difference A - B of two vectors A and B to be the vector sum of A and -B:
A - B = A + (-B)
A
vector quantity such as a displacement can be multiplied by a scalar quantity (an
ordinary number). The displacement 2A is a displacement
(vector quantity) in the same direction as the vector A but twice as long; this is the same as adding A to itself. In general, when a vector A is multiplied by a scalar c, the result cA has
magnitude |c|A (the absolute value of c multiplied by the magnitude
of the vector A). If c is positive, cA is in the same
direction as A; if c is negative,
cA is in the direction opposite to A.
