NDA Math Three Dimensional Geometry

The vector joining P and Q will be perpendicular to the direction vector of the line. Hence, the dot product of the vector joining P and Q and the direction vector of the line will be zero.
Let

x

2

=

y−2

3

=

z−3

4

=t
x=2t
y=3t+2
z=4t+3
Hence,
∴4t−6+9t+9+16t−32=0
29t−29=0
t=1
=(−1)

∧

i

+(6)

∧

j

−(4)

∧

k

Therefore the length of the line segment PQ will be given by magnitude of the vector joining P and Q.
Hence,
|P

→

Q

|=√1+36+16
=√53 units