It is given that
Line joining O (0, 0, 0) and P (2, 3, 4) is written as
OP = 2i + 3j + 4k
Line joining O (0, 0, 0) and Q (1, -2, 1) is written as
OQ = i – 2j + k
In order prove that these two lines are perpendicular we must show that angle between these two lines is π/2
Dot product
OP. OQ = 0
Substituting the values
(2i + 3j + 4k). (i – 2j + k) = 2 – 6 + 4 = 0
Therefore, it is proved that these two lines are perpendicular.