Multi variable Question. a) Give the vector and linear equations of the plane co
ID: 2856124 • Letter: M
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Multi variable Question. a) Give the vector and linear equations of the plane containing the point (1,2,3) and the line with parametric equations: x(t)=4-2t Y(t)=3+5t. z(t)=7+4t b) Determine whether the line through the points (4,1,-1) and (2,5,3) is perpendicular to the line through the points (6,-3,-5) and (5,1,4) MAKE SURE YOU SHOW ALL WORK WITH NOTATION PLEASE. Multi variable Question. a) Give the vector and linear equations of the plane containing the point (1,2,3) and the line with parametric equations: x(t)=4-2t Y(t)=3+5t. z(t)=7+4t b) Determine whether the line through the points (4,1,-1) and (2,5,3) is perpendicular to the line through the points (6,-3,-5) and (5,1,4) MAKE SURE YOU SHOW ALL WORK WITH NOTATION PLEASE. a) Give the vector and linear equations of the plane containing the point (1,2,3) and the line with parametric equations: x(t)=4-2t Y(t)=3+5t. z(t)=7+4t b) Determine whether the line through the points (4,1,-1) and (2,5,3) is perpendicular to the line through the points (6,-3,-5) and (5,1,4) MAKE SURE YOU SHOW ALL WORK WITH NOTATION PLEASE.Explanation / Answer
a) Give the vector and linear equations of the plane containing the point (1,2,3) and the line with parametric equations: x(t)=4-2t Y(t)=3+5t. z(t)=7+4t
let given point be P(1,2,3), general point on plane X(x,y,z)
PX=<x-1,y-2,z-3>
for t=0,point on line is A(4,3,7), t=1, point is B(2,8,11)
PA=A-P=(4,3,7)-(1,2,3)=<3,1,4>
PB=B-P=(2,8,11)-(1,2,3)=<1,6,8>
PA x PB =<(1*8)-(6*4) ,(1*4)-(3*8) ,(3*6)-(1*1)>
PA x PB =<-16 ,-20 ,17>
vector equation of plane is (PA x PB).PX=0
vector equation of plane is <-16 ,-20 ,17>.<x-1,y-2,z-3>=0
-16(x-1)-20(y-2)+17(z-3)=0
-16x+16-20y+40+17z-51=0
16x+20y-17z=5 is linear equation of plane
b) Determine whether the line through the points (4,1,-1) and (2,5,3) is perpendicular to the line through the points (6,-3,-5) and (5,1,4)
let points be A(4,1,-1) ,B(2,5,3) ,C(6,-3,-5) and D(5,1,4)
AB=B-A=(2,5,3) -(4,1,-1) =<-2,4,4>
CD=D-C=(5,1,4)-(6,-3,-5)=<-1,4,9>
AB.CD =<-2,4,4>.<-1,4,9>
AB.CD =(-2*-1)+(4*4)+(4*9)
AB.CD =54 not equal to zero
so lines are not perpendicular
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