AP EAPCET 25-Aug-2021 Shift 2 Paper

Let OA=2

^

i

+3

^

j

−

^

k

=a
OB=3

^

i

+2

^

j

+

^

k

=b
OC=

^

i

+

^

j

+4

^

k

=c
∵ Vector equation of a plane passing through three points with position vectors a, b and c is
(r−a).[(b−a)×(c−a)]=0 ...(i)
where r=x

^

i

+y

^

j

+z

^

k

Let us find b−a=(3

^

i

+2

^

j

+

^

k

)−(2

^

i

+3

^

j

−

^

k

)
=

^

i

−

^

j

+2

^

k

c−a=(

^

i

+

^

j

+4

^

k

)−(2

^

i

+3

^

j

−

^

k

)
=−

^

i

−2

^

j

+5

^

k

∴ (b−a)×(c−a)=|

^ i	^ j	^ k
1	−1	2
−1	−2	5

|
=

^

i

(−5+4)−

^

j

(5+2+

^

k

(−2−1)
=−

^

i

−7

^

j

−3

^

k

and r−a=(x−2)

^

i

+(y−3

^

j

+(z+1)

^

k

From Eq. (i), we get
−(x−2)−7(y−3)−3(z+1)=0
−x−7y−3z+2+21−3=0
x+7y+3z=20
Since, (2,−3,13) and (2,

3

2

,

5

2

) satisfies above equation.
Hence, its position vectors are 2

^

i

−3

^

j

+13

^

k

and 2

^

i

+

3

2

^

j

+

5

2

^

k