In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(a) (0,0,0) 3x-4y+12z=3
(b) (3,-2,1) 2x-y+2z+3=0
(c) (2,3,-5) x+2y-2z=9
(d) (-6,0,0) 2x-3y+6z-2=0
It is known that the distance between a point P(x1,y1,z1)and a plane Ax-+By+Cz=D, is given by,
d=\(\begin{vmatrix}\frac{AX_1+By_1+Cz_1-D}{\sqrt{A^2+B^2+C^2}} \end{vmatrix}\)|...(1)
(a) The given point is (0,0,0) and the plane is 3x-4y+12z=3
∴d=\(\begin{vmatrix}\frac{3*0-4*0+12*0-3}{\sqrt{(3)^2+(-4)^2+(12)^2}} \end{vmatrix}\)
=\(\frac{3}{\sqrt{169}}\)=\(\frac{3}{13}\)
(b)The given point is (3,-2,1) and the plane is 2x-y+2z+3=0
∴d=\(\begin{vmatrix}\frac{2*3-(-2)+2*1+3}{\sqrt{(2)^2+(-1)^2+(2)^2}} \end{vmatrix}\)
=|\(\frac{13}{3}\)| =\(\frac{13}{3}\)
(c)The given point is (2,3,-5) and the plane is x+2y-2z=9
∴d=\(\begin{vmatrix}\frac{2+2*3-2(-5)-9}{\sqrt{(1)^2+(2)^2+(-2)^2}} \end{vmatrix}\)
=\(\frac{9}{3}\)
=3
(d)The given point is (-6,0,0) and the plane is 2x-3y+6z-2=0
∴d=\(\begin{vmatrix}\frac{2(-6)-3*0+6*0-2}{\sqrt{(2)^2+(-3)^2+(6)^2}} \end{vmatrix}\)
=\(\begin{vmatrix}\frac{-14}{\sqrt{49}} \end{vmatrix}\)
=\(\frac{14}{7}\)
=2
List - I | List - II | ||
(P) | γ equals | (1) | \(-\hat{i}-\hat{j}+\hat{k}\) |
(Q) | A possible choice for \(\hat{n}\) is | (2) | \(\sqrt{\frac{3}{2}}\) |
(R) | \(\overrightarrow{OR_1}\) equals | (3) | 1 |
(S) | A possible value of \(\overrightarrow{OR_1}.\hat{n}\) is | (4) | \(\frac{1}{\sqrt6}\hat{i}-\frac{2}{\sqrt6}\hat{j}+\frac{1}{\sqrt6}\hat{k}\) |
(5) | \(\sqrt{\frac{2}{3}}\) |
The number of diagonals of a polygon is 35. If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having AB as one of its sides is:
What is the Planning Process?
The shortest perpendicular distance from the point to the given plane is the distance between point and plane. In simple terms, the shortest distance from a point to a plane is the length of the perpendicular parallel to the normal vector dropped from the particular point to the particular plane. Let's see the formula for the distance between point and plane.
Read More: Distance Between Two Points