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CH 1342 Final Exam, Fall 2023 Find a standard equation for the plane containing the line vec(L)(t)=(-4,0,6)+t(3,5,1) and the point (2,1,3)

CH 13421342\newlineFinal Exam, Fall 20232023\newlineFind a standard equation for the plane containing the line L⃗(t)=(−4,0,6)+t(3,5,1) \vec{L}(t)=(-4,0,6)+t(3,5,1) and the point (2,1,3) (2,1,3)

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Q. CH 13421342\newlineFinal Exam, Fall 20232023\newlineFind a standard equation for the plane containing the line L⃗(t)=(−4,0,6)+t(3,5,1) \vec{L}(t)=(-4,0,6)+t(3,5,1) and the point (2,1,3) (2,1,3)
  1. Identify Point on Line: Identify a point on the line L⃗(t)\vec{L}(t) by setting t=0t=0.\newlinePoint on the line: (−4,0,6)(-4,0,6).
  2. Find Direction Vector: Find the direction vector of the line L⃗(t)\vec{L}(t) which is also a vector parallel to the plane.\newlineDirection vector: (3,5,1)(3,5,1).
  3. Find Connecting Vector: Find the vector connecting the point on the line to the given point (2,1,3)(2,1,3).\newlineConnecting vector: (2−(−4),1−0,3−6)=(6,1,−3)(2 - (-4), 1 - 0, 3 - 6) = (6,1,-3).
  4. Calculate Normal Vector: Calculate the normal vector of the plane by taking the cross product of the direction vector and the connecting vector.\newlineNormal vector: (5∗(−3)−1∗1,1∗3−(−4)∗(−3),(−4)∗1−6∗3)=(−15−1,3−12,−4−18)(5*(-3) - 1*1, 1*3 - (-4)*(-3), (-4)*1 - 6*3) = (-15 - 1, 3 - 12, -4 - 18).
  5. Simplify Normal Vector: Simplify the normal vector.\newlineNormal vector: (−16,−9,−22)(-16, -9, -22).
  6. Write Plane Equation: Use the normal vector and the point on the line to write the standard equation of the plane.\newlineEquation: −16(x+4)−9(y−0)−22(z−6)=0-16(x + 4) - 9(y - 0) - 22(z - 6) = 0.
  7. Expand Equation: Expand and simplify the equation of the plane.\newlineEquation: −16x−64−9y−22z+132=0-16x - 64 - 9y - 22z + 132 = 0.
  8. Combine Like Terms: Combine like terms to get the final standard equation of the plane.\newlineEquation: −16x−9y−22z+68=0-16x - 9y - 22z + 68 = 0.

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