only do part a & f 13. In each case find the equation of the plane: a) Passing t

Question

only do part a & f

13. In each case find the equation of the plane: a) Passing through A(3, 1, 2), B(5, 1, 3), and C(-4, 2, 0) b) Passing through A(6, 1, 1), B(1, 0, 0), and C(21, 3, 2) c) Passing through A(6, 1, 0) and perpendicular to the 1 4 line y zl d) Passing through A(1 1, 3) and perpendicular to the line [x y Z1 15 5 5] t 8 2 0] e) Passing through PO2, 0, 3) and parallel to the plane through the points A(1, 1, 5), B(0, 1, -2), and C(1, 0, 6) f) Passing through PO2 1, 5) and parallel to the plane through the points A(3, -7, 1), B(2, 0, 1), and C(1, 3, 0)

Explanation / Answer

a.)

Plane passes through A(3,1,2),B(5,-1,3) and C(-4,2,0)

vectors AB and AC = (B-A) and (C-A) = (2,-2,1) and (-7,1,-2)

normal vector = cross product of AB and AC = (2,-2,1) x (-7,1,-2) = (3,-3,-12)

Plane equation = (3,-3,-12).(x,y,z)+d = 0

or

3x-3y-12z+d=0

Substitute (x,y,z) = (3,1,2) to get value of d=-3x+3y+12z = -9+3+24 = 18

Equation of plane: 3x-3y+12z+18=0 or x-y+4z+6=0

f.)

Plane passes through P(2,-1,5) and is parallel to plane through A(3,-7,1),B(2,0,-1) and C(1,3,0)

vectors AB and AC = (B-A) and (C-A) = (-1,7,-2) and (-2,10,-1)

normal vector = cross product of AB and AC = (-1,7,-2) x (-2,10,-1)= (13,3,4)

Plane equation = (13,3,4).(x,y,z)+d = 0

or

13x+3y+3z+d=0

Substitute (x,y,z) = (2,-1,5) to get value of d=-13x-3y-3z = -26+3-15=-38

Equation of plane: 13x+3y+3z-38=0

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