I need help with these three questions please: 1. Find the general equation for
ID: 2937347 • Letter: I
Question
I need help with these three questions please:1. Find the general equation for the plane P1 that hasnormal n1= (1,1,1) and passses through the point (1,2,3).
2. Find the general equation of the plane P2 that passesthrough the points (0,0,0) (0,1,2) and (2,2,2).
3. Find the parametric equations for the line ofintersection of the planes P1 and P2 and determine a vector Dparallel to the line.
1. Find the general equation for the plane P1 that hasnormal n1= (1,1,1) and passses through the point (1,2,3).
2. Find the general equation of the plane P2 that passesthrough the points (0,0,0) (0,1,2) and (2,2,2).
3. Find the parametric equations for the line ofintersection of the planes P1 and P2 and determine a vector Dparallel to the line.
Explanation / Answer
QuestionDetails:1. Find the general equation for the plane P1 that hasnormal n1= (1,1,1)=[A,B,C]
and passses through the point (1,2,3).[X1,X2,X3] FORMULA
A(X-X1)+B(Y-Y1)+C(Z-Z1)=0........................1
X-1+Y-2+Z-3=0
X+Y+Z=6.........................2
2. Find the general equation of the plane P2 that passesthrough the points
[X1,Y1,Z1]= (0,0,0) ; (0,1,2) and (2,2,2). FROM FORMULA 1
AX+BY+CZ=0....................................3
(0,1,2) IS A POINT ON THE PLANE .SO
1B+2C=0...........B=-2C............................4
(2,2,2) IS A POINT ON THE PLANE.SO
2A+2B+2C=0......A+B+C=0......A-2C+C=0.......A=C..............5
PUTTING 4 AND 5 IN 3, WE GET
CX-2CY+CZ=0....OR
EQN. OF PLANE IS
X-2Y+Z=0......................................6
3. Find the parametric equations for the line ofintersection of the planes
P1 and P2 and determine a vector D parallel to the line.
P1 IS GIVEN BY
X+Y+Z=6....................................2
P2 IS GIVEN BY
X-2Y+Z=0................................................6
HENCE PARAMETRIC EQN. OF LINE THROUGH THEIR INTERSECTION IS
X+Y+Z-6+K(X-2Y+Z)=0
X(1+K)+Y(1-2K)+Z(1+K)=0 IS THE EQN. OF LINE FOR ANY REAL VALUE OFK
ITS DRS ARE
[(1+K);(1-2K);(1+K)]
LINE D PARALLEL TO IT WILL HAVE SAME DRS
VECTOR IS [(1+K);(1-2K);(1+K)] IN GENERAL FOR ANY REAL VALUE OFK
OR IN PARTICULAR SAY TAKING K=1 WE GET
[2,-1,2]