Sometimes x, y and z are also termed as rectangular
components The length of any vector 
ˆ
ˆ
ˆ
r
xi
yj
zk
=
+
+
r
, is readily determined by applying the
Pythagoras theorem twice We note that in the right angle triangle OQP1 (Fig 10 14)
| OP |1
uuuur  =
2
2
2
2
|OQ| +|QP |1
x
y
=
+
uuuur
uuur
,
and in the right angle triangle OP1P, we have
OP
uuur  =
2
2
2
2
2
1
1
| OP |
| P P |
(
)
x
y
z
 
 
 
 
uuur
uuur
Hence, the length of any vector 
ˆ
ˆ
ˆ+
r
xi
yj
zk
=
+
r
 is given by
|
|
rr  =
2
2
2
ˆ
ˆ
ˆ
|
| =
xi
yj
zk
x
y
z
+
+
+
+
If  and 
a
rb
r
 are any two vectors given in the component form 
1
2
3 ˆ
ˆ
ˆ+
a i
a j
a k
+
 and
1
2
3 ˆ
ˆ
ˆ
b i
b j
b k
+
+
, respectively, then
(i)
the sum (or resultant) of the vectors 
aand 
rb
r
 is given by
a
b
+
r
r
 =
1
1
2
2
3
3 ˆ
ˆ
ˆ
(
)
(
)
(
)
a
b i
a
b
j
a
b k
+
+
+
+
+
(ii)
the difference of the vector 
aand 
rb
r
 is given by
a
−b
r
r
=
1
1
2
2
3
3 ˆ
ˆ
ˆ
(
)
(
)
(
)
a
b i
a
b
j
a
b k
−
+
−
+
−
(iii)
the vectors 
aand 
rb
r
 are equal if and only if
a1 = b1, a2 = b2   and   a3 = b3
(iv)
the multiplication of vector ar  by any scalar λ is given by
a
λr  =
1
2
3 ˆ
ˆ
ˆ
(
)
(
)
(
)
a i
a
j
a k
 
 
 
 
 
© NCERT
not to be republished
VECTOR ALGEBRA
435
The addition of vectors and the multiplication of a vector by a scalar together give
the following distributive laws:
Let 
a and 
rb
r
 be any two vectors, and k and m be any scalars