Page 85 - Class-11-Physics-Part-1_Neat

MOTION IN A PLANE 71

So far we have considered a vector lying in
B = B x i + B y j
an x-y plane. The same procedure can be used
to resolve a general vector A into three Let R be their sum. We have
components along x-, y-, and z-axes in three R = A + B
dimensions. If α, β, and γ are the angles*
=
+
( A i + A j ) ( B i + B ) j (4.19a)
between A and the x-, y-, and z-axes, respectively x y x y
[Fig. 4.9(d)], we have Since vectors obey the commutative and
associative laws, we can arrange and regroup
the vectors in Eq. (4.19a) as convenient to us :

i +
R = (A x + B x ) ( A y + B y ) j (4.19b)

SinceR = R x i + R y j (4.20)

we have, R = A + B , R = A + B (4.21)
x
x
x
y
y
y
Thus, each component of the resultant
vector R is the sum of the corresponding
components of A and B.
In three dimensions, we have

A = A x i + A y j + A z k

B = B x i + B y j + B z k
(d)

R = A + B = R i + R j + R k
Fig. 4.9 (d) A vector A resolved into components along x y z
x-, y-, and z-axes
with R = A + B x
x
x
A = A cos , A = A cos , A = A cos (4.16a) R = A + B y
γ
α
β
y
y
y
x
z
In general, we have R = A + B z (4.22)
z
z
ˆ
ˆ
A = A i ˆ + A j + A k (4.16b) This method can be extended to addition and
x y z
The magnitude of vector A is subtraction of any number of vectors. For
A = A + A + A 2 (4.16c) example, if vectors a, b and c are given as
2
2
x y z
A position vector r can be expressed as a = a i + a j + a k
x y z
r = x i + y j + z k (4.17)

b = b x i + b y j + b z k
where x, y, and z are the components of r along
x-, y-, z-axes, respectively. c = c i + c j + c k (4.23a)
x y z
4.6 VECTOR ADDITION – ANALYTICAL then, a vector T = a + b – c has components :
METHOD T = a + b − c
x x x x
Although the graphical method of adding vectors T = a + b − c (4.23b)
y y y y
helps us in visualising the vectors and the T = a + b − c .
resultant vector, it is sometimes tedious and has z z z z
limited accuracy. It is much easier to add vectors
by combining their respective components. t Example 4.2 Find the magnitude and
direction of the resultant of two vectors A
Consider two vectors A and B in x-y plane with
and B in terms of their magnitudes and
components A , A and B , B :
x y x y angle θ between them.

A = A i + A j (4.18)
x y
* Note that angles α, β, and γ are angles in space. They are between pairs of lines, which are not coplanar.
2018-19