Question 34516: A straight line L and a plane P in R3 are defined by the equations 2x − 3y + z = 1 and (x, y, z) = (1,−1, 2) + t (1, 2, 3), t any real number.
(a) Say briefly but clearly how you know the line L does not pass through the point
(−1,−5,−3).
(b) Say briefly but clearly how you know that the plane P does not pass through the origin.
(c) Write the equation for the line perpendicular to the plane P and passing through the
origin.
(d) Write the equation for the plane perpendicular to L and passing through the point
(0, 1, 3).
(e) Write the equation for the xz-plane.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! SEE ANSWERS BELOW
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A straight line L and a plane P in R3 are defined by the equations
EQN.OF PLANE P........ 2x − 3y + z = 1
and EQN.OF LINE L.....(x, y, z) = (1,−1, 2) + t (1, 2, 3), t any real number.
(a) Say briefly but clearly how you know the line L does not pass through the point
(−1,−5,−3)....THIS DOES NOT SATISFY EQN.L.FOR ONE VALUE OF T AS SHOWN BELOW
-1=1+T...OR....T=-2...
-5=-1+2T...OR....T=-2..OK
-3=2+3T.....OR....T=-5/3....NOT TALLYING
(b) Say briefly but clearly how you know that the plane P does not pass through the origin....ORIGIN IS (0,0,0)...THIS DOES NOT SATISFY EQN.P...WE GET 0=1
(c) Write the equation for the line perpendicular to the plane P and passing through the
origin....LINE PERPENDICULAR TO PLANE IS PARALLEL TO PLANE'S NORMAL WHICH HAS DRS OF 2,-3,1...LINE PASSES THROUGH (0,0,0)..SO EQN OF LINE IS
X/2 = Y/-3 = Z/1
(d) Write the equation for the plane perpendicular to L and passing through the point
(0, 1, 3)..........DRS OF LINE ARE 1,2,3...THE REQD.PLANE IS PERPENDICULAR TO THE LINE.HENCE ITS NORMAL IS PARALLEL TO LINE.HENCE DRS OF NORMAL ARE SAME AS DRS OF LINE..THAT IS 1,2,3...HENCE EQN.OF PLANE IS X+2Y+3Z=K...IT PASSES THROUGH
(0,1,3)....SO.....0+1*2+3*3=K=11
EQN. OF REQD. PLANE IS ..................X+2Y+3Z=11
(e) Write the equation for the xz-plane.
Y=0
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