In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z=2 (b) x+y+z=1
(c)2x+3y-z=5 (d) 5y+8=0
(a) The equation of the plane is z=2 or 0x+0y+z=2...(1)
The direction ratios of the normal are 0, 0, and 1.
∴
Dividing both sides of equation(1) by 1, we obtain
0.x+0.y+1.z=2
This is of the form lx+my+nz=d, where l, m, n are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.
Therefore, the direction cosines are 0, 0, and 1 and the distance of the plane from the origin is 2 units.
(b) x+y+z=1...(1)
The direction ratios of normal are 1, 1, and 1.
∴
Dividing both sides of equation(1) by , we obtain
...(2)
This equation is the form lx+my+nz=d, where l, m, n are the direction cosines of normal to the plane and d is the distance of normal from the origin.
Therefore, the direction cosines of the normal are , and and the distance of normal from the origin is units.
(c) 2x+3y-z=5...(1)
The direction ratios of normal are 2, 3, and -1.
∴
Dividing both sides of equation(1) by 14, we obtain
This equation is of the form lx+my+nz=d, where l, m, n are the direction cosines of normal to the plane and d is the distance of normal from the origin.
Therefore, the direction cosines of the normal to the plane are , and and the distance of normal from the origin is units.
(d) 5y+8=0
0x-5y+0z=8...(1)
The direction ratios of normal are 0, -5, and 0.
∴ =5
Dividing both sides of equation(1) by 5, we obtain
-y=
This equation is of the form lx+my+nz=d, where l, m, n are the direction cosines of normal to the plane and d is the distance of normal from the origin.
Therefore, the direction cosines of the normal to the plane are 0, -1, and 0 and the distance of normal from the origin is units.
List - I | List - II | ||
(P) | γ equals | (1) | |
(Q) | A possible choice for is | (2) | |
(R) | equals | (3) | 1 |
(S) | A possible value of is | (4) | |
(5) |
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely: