UPSC NDA Math Model Paper 3

Given the origin (0, 0, 0) and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are co-planer
-⇒

→

a

=

→

OR

=(x,y,z)
⇒

→

b

=

→

OP

=(2,3,4)
⇒

→

c

=

→

OQ

=(1,2,3)
Here,

→

a

,

→

b

and

→

c

are co planer
The three vectors are coplanar if their scalar triple product is zero..
⇒

→

a

.(

→

b

×

→

c

)=0
⇒|

x	y	z
2	3	4
1	2	3

|=0
⇒x(9−8)−y(6−4)+z(4−3)=0
⇒x−2y+z=0
Hence, if the origin and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are co-planar then x - 2y + z = 0