Question

The distance of the point \[2i + j - k\] from the plane \[r.\left( {i - 2j + 4k} \right) = 9\;\;\] is
A) \[\dfrac{{13}}{{\sqrt {21} }}\]
B) \[\dfrac{{13}}{{21}}\]
C) \[\dfrac{{13}}{{3\sqrt {21} }}\]

Answer 1

Hint: in this question we have to find the distance of a point from a given plane. It can be found by simply applying the formula of distance of a point from the plane.

Formula Used: Distance of a given point with position vector a from plane is given by
\[\dfrac{{\left| {a.n - d} \right|}}{{\left| n \right|}}\]
Where
a is position vector of point
n is vector normal to the given plane
d is any constant

Complete step by step solution: Given equation of a plane is
\[r.\left( {i - 2j + 4k} \right) = 9\;\;\]
It is in the form of
\[r.n = d\]
Compare this equation with equation of given plane
\[n = i - 2j + 4k\]
\[d = 9\]
Now, position vector of point is \[2i + j - k\]
\[a = 2i + j - k\]
Now required distance is given by
\[\dfrac{{\left| {a.n - d} \right|}}{{\left| n \right|}}\]
Where
a is position vector of point
n is vector normal to the given plane
d is any constant

\[\dfrac{{\left| {(2i + j - k).(i - 2j + 4k) - 9} \right|}}{{\sqrt {1 + 4 + 16} }}\]
\[\dfrac{{\left| {2 - 2 - 4 - 9} \right|}}{{\sqrt {21} }}\]
Distance of a point from given plane is
\[\dfrac{{13}}{{\sqrt {21} }}\]

Option ‘A’ is correct
Note: Here we need to remember the formula of perpendicular distance of a point with given position vector from the given plane is\[\dfrac{{\left| {a.n - d} \right|}}{{\left| n \right|}}\]. Then put the value of all parameters to get distance.