If 
and 
then express 
in the form of 
where 
is parallel to 
and 
is perpendicular to 
There are two vectors 
= 2
- 2
and 
= 7
+ 2
- 3
and now we have to express 
as the sum of two vectors 
1 and 
2 such that 
1 is parallel to 
and 
2 is perpendicular to 
.
Let 
1 = p
+ q
+ r
and 
2 = x
+ y
+ z
So, 
1 = λ
because both the vectors are parallel to each other and 
2.
= 0 because both the vectors are perpendicular to each.
This is the sum equation.
Taking dot product of 
on both sides, we get,
18 = 2p – q – 2r,
18 = 4λ + λ + 4λ,
λ = 2.
So, 
= 4
- 2
- 4
= 3
+ 4
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