Question

Find the image of the point having position vector `hat"i" + 3hat"j" + 4hat"k"` in the plane `hat"r" * (2hat"i" - hat"j" + hat"k") + 3` = 0.

Answer 1

Let the given point be `"P"(hat"i" + 3hat"j" + 4hat"k")` and Q be the image of P in the plane

`hat"r" * (2hat"i" - hat"j" + hat"k") + 3` = 0 as shown in the figure

Then PQ is the normal to the plane.

Since PQ passes through P and is normal to the given plane

So the equation of PQ is given by `vec"r" = (hat"i" + 3hat"j" + 4hat"k") + lambda(2hat"i" - hat"j" + hat"k")`

Since Q lies on the line PQ

The position vector of Q can be expressed as

`(hat"i" + 3hat"j" + 4hat"k") + lambda(2hat"i" - hat"j" + hat"k")`

i.e., `(1 + 2lambda)hat"i" + (3 - lambda)hat"j" + (4 + lambda)hat"k"`

Since R is the mid point of PQ, the position vector of R is

`([(1 + 2lambda)hat"i" + (3 - lambda)hat"j" + (4 + lambda)hat"k"] + [hat"i" + 3hat"j" + 4hat"k"])/2`

i.e., `(lambda + 1)hat"i" + (3 - lambda/2)hat"j" + (4 + lambda/2)hat"k"`

Again, since R lies on the plane `vec"r" * (2hat"i" - hat"j" + hat"k") + 3` = 0, we have

`{(lambda + 1)hat"i" + (3 - lambda/2)hat"j" + (4 + lambda/2)hat"k"} * (2hat"i" - hat"j" + hat"k") + 3` = 0

⇒ λ = –2

Hence, the position vector of Q is `(hat"i" + 3hat"j" + 4hat"k") -2(2hat"i" - hat"j" + hat"k")`

i.e. `-3hat"i" + 5hat"j" + 2hat"k"`.