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Equation of a plane practice problems

A plane through three points, from a normal vector. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 9.31

hard
Find an equation of the plane through the points \( \displaystyle (1, 2, 3),\ (-1, -1, 3),\ (-1, 3, 0) \).
  1. \[ \left[\begin{matrix}-2\\-3\\0\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-2\\1\\-3\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}9\\-6\\-8\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -27 \]
    n · P gives the constant.Reviewed
  5. The plane is 9*x - 6*y - 8*z = -27.
    Reviewed
Answer \( 9 x - 6 y - 8 z = -27 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.32

hard
Find an equation of the plane through the points \( \displaystyle (1, -1, 1),\ (-2, 1, 2),\ (0, -2, -3) \).
  1. \[ \left[\begin{matrix}-3\\2\\1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-1\\-1\\-4\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-7\\-13\\5\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 11 \]
    n · P gives the constant.Reviewed
  5. The plane is -7*x - 13*y + 5*z = 11.
    Reviewed
Answer \( - 7 x - 13 y + 5 z = 11 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.35

hard
Find an equation of the plane through the points \( \displaystyle (-3, -1, 1),\ (-1, 0, 0),\ (0, 0, -3) \).
  1. \[ \left[\begin{matrix}2\\1\\-1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}3\\1\\-4\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-3\\5\\-1\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 3 \]
    n · P gives the constant.Reviewed
  5. The plane is -3*x + 5*y - z = 3.
    Reviewed
Answer \( - 3 x + 5 y - z = 3 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.36

hard
Find an equation of the plane through the points \( \displaystyle (1, -1, 3),\ (3, 1, 0),\ (2, 1, 3) \).
  1. \[ \left[\begin{matrix}2\\2\\-3\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}1\\2\\0\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}6\\-3\\2\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 15 \]
    n · P gives the constant.Reviewed
  5. The plane is 6*x - 3*y + 2*z = 15.
    Reviewed
Answer \( 6 x - 3 y + 2 z = 15 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.38

hard
Find an equation of the plane through the points \( \displaystyle (-2, -2, -1),\ (1, -1, -1),\ (-1, -2, -3) \).
  1. \[ \left[\begin{matrix}3\\1\\0\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}1\\0\\-2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-2\\6\\-1\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -7 \]
    n · P gives the constant.Reviewed
  5. The plane is -2*x + 6*y - z = -7.
    Reviewed
Answer \( - 2 x + 6 y - z = -7 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.41

hard
Find an equation of the plane through the points \( \displaystyle (3, 3, -3),\ (-2, 2, -3),\ (-2, -3, -1) \).
  1. \[ \left[\begin{matrix}-5\\-1\\0\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-5\\-6\\2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-2\\10\\25\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -51 \]
    n · P gives the constant.Reviewed
  5. The plane is -2*x + 10*y + 25*z = -51.
    Reviewed
Answer \( - 2 x + 10 y + 25 z = -51 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.43

hard
Find an equation of the plane through the points \( \displaystyle (2, 0, -1),\ (-2, 1, 2),\ (3, -2, -2) \).
  1. \[ \left[\begin{matrix}-4\\1\\3\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}1\\-2\\-1\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}5\\-1\\7\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 3 \]
    n · P gives the constant.Reviewed
  5. The plane is 5*x - y + 7*z = 3.
    Reviewed
Answer \( 5 x - y + 7 z = 3 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.77

hard
Find an equation of the plane through the points \( \displaystyle (-3, 0, 0),\ (-3, 3, -2),\ (3, -2, 2) \).
  1. \[ \left[\begin{matrix}0\\3\\-2\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}6\\-2\\2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}1\\-6\\-9\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -3 \]
    n · P gives the constant.Reviewed
  5. The plane is x - 6*y - 9*z = -3.
    Reviewed
Answer \( x - 6 y - 9 z = -3 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.78

hard
Find an equation of the plane through the points \( \displaystyle (0, 1, 0),\ (-2, -1, 3),\ (3, -1, -2) \).
  1. \[ \left[\begin{matrix}-2\\-2\\3\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}3\\-2\\-2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}2\\1\\2\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 1 \]
    n · P gives the constant.Reviewed
  5. The plane is 2*x + y + 2*z = 1.
    Reviewed
Answer \( 2 x + y + 2 z = 1 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.79

hard
Find an equation of the plane through the points \( \displaystyle (-1, 2, 2),\ (-3, -1, 3),\ (2, -2, -2) \).
  1. \[ \left[\begin{matrix}-2\\-3\\1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}3\\-4\\-4\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}16\\-5\\17\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 8 \]
    n · P gives the constant.Reviewed
  5. The plane is 16*x - 5*y + 17*z = 8.
    Reviewed
Answer \( 16 x - 5 y + 17 z = 8 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.84

hard
Find an equation of the plane through the points \( \displaystyle (-1, 0, 0),\ (0, -3, -1),\ (-2, 0, -1) \).
  1. \[ \left[\begin{matrix}1\\-3\\-1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-1\\0\\-1\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}3\\2\\-3\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -3 \]
    n · P gives the constant.Reviewed
  5. The plane is 3*x + 2*y - 3*z = -3.
    Reviewed
Answer \( 3 x + 2 y - 3 z = -3 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.85

hard
Find an equation of the plane through the points \( \displaystyle (1, 1, 2),\ (0, 3, 3),\ (1, -2, 3) \).
  1. \[ \left[\begin{matrix}-1\\2\\1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}0\\-3\\1\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}5\\1\\3\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 12 \]
    n · P gives the constant.Reviewed
  5. The plane is 5*x + y + 3*z = 12.
    Reviewed
Answer \( 5 x + y + 3 z = 12 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.86

hard
Find an equation of the plane through the points \( \displaystyle (-3, 3, -3),\ (1, 2, -2),\ (2, -3, 1) \).
  1. \[ \left[\begin{matrix}4\\-1\\1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}5\\-6\\4\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}2\\-11\\-19\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 18 \]
    n · P gives the constant.Reviewed
  5. The plane is 2*x - 11*y - 19*z = 18.
    Reviewed
Answer \( 2 x - 11 y - 19 z = 18 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.89

hard
Find an equation of the plane through the points \( \displaystyle (2, -2, -2),\ (-2, -1, -2),\ (1, -3, 1) \).
  1. \[ \left[\begin{matrix}-4\\1\\0\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-1\\-1\\3\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}3\\12\\5\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -28 \]
    n · P gives the constant.Reviewed
  5. The plane is 3*x + 12*y + 5*z = -28.
    Reviewed
Answer \( 3 x + 12 y + 5 z = -28 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.90

hard
Find an equation of the plane through the points \( \displaystyle (-1, 3, -1),\ (1, 1, 0),\ (0, -3, -3) \).
  1. \[ \left[\begin{matrix}2\\-2\\1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}1\\-6\\-2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}2\\1\\-2\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 3 \]
    n · P gives the constant.Reviewed
  5. The plane is 2*x + y - 2*z = 3.
    Reviewed
Answer \( 2 x + y - 2 z = 3 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.108

hard
Find an equation of the plane through the points \( \displaystyle (0, -2, -1),\ (2, 0, 0),\ (-1, 1, -2) \).
  1. \[ \left[\begin{matrix}2\\2\\1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-1\\3\\-1\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-5\\1\\8\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -10 \]
    n · P gives the constant.Reviewed
  5. The plane is -5*x + y + 8*z = -10.
    Reviewed
Answer \( - 5 x + y + 8 z = -10 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.109

hard
Find an equation of the plane through the points \( \displaystyle (-3, 2, 1),\ (3, -2, 0),\ (-3, 3, 3) \).
  1. \[ \left[\begin{matrix}6\\-4\\-1\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}0\\1\\2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-7\\-12\\6\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 3 \]
    n · P gives the constant.Reviewed
  5. The plane is -7*x - 12*y + 6*z = 3.
    Reviewed
Answer \( - 7 x - 12 y + 6 z = 3 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.112

hard
Find an equation of the plane through the points \( \displaystyle (2, 0, 2),\ (-1, -1, 2),\ (0, 1, 0) \).
  1. \[ \left[\begin{matrix}-3\\-1\\0\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}-2\\1\\-2\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}2\\-6\\-5\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -6 \]
    n · P gives the constant.Reviewed
  5. The plane is 2*x - 6*y - 5*z = -6.
    Reviewed
Answer \( 2 x - 6 y - 5 z = -6 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.132

hard
Find an equation of the plane through the points \( \displaystyle (1, -1, 2),\ (-1, -2, -2),\ (2, -3, 2) \).
  1. \[ \left[\begin{matrix}-2\\-1\\-4\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}1\\-2\\0\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-8\\-4\\5\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ 6 \]
    n · P gives the constant.Reviewed
  5. The plane is -8*x - 4*y + 5*z = 6.
    Reviewed
Answer \( - 8 x - 4 y + 5 z = 6 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.133

hard
Find an equation of the plane through the points \( \displaystyle (1, -2, -3),\ (-3, -2, 3),\ (2, 2, 1) \).
  1. \[ \left[\begin{matrix}-4\\0\\6\end{matrix}\right] \]
    PQ.Reviewed
  2. \[ \left[\begin{matrix}1\\4\\4\end{matrix}\right] \]
    PR.Reviewed
  3. \[ \left[\begin{matrix}-12\\11\\-8\end{matrix}\right] \]
    A normal vector n = PQ × PR (scaled).Reviewed
  4. \[ -10 \]
    n · P gives the constant.Reviewed
  5. The plane is -12*x + 11*y - 8*z = -10.
    Reviewed
Answer \( - 12 x + 11 y - 8 z = -10 \)

✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More equation of a plane practice

Find an equation of the plane through the points \( \displaystyle (2, 1, -1),\ (-3, -3, 0),\ (2, -1, -1) \).
Problem 9.33medium✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-2, -1, -2),\ (2, 0, 3),\ (0, 0, 1) \).
Problem 9.40mediumNot checked
Find an equation of the plane through the points \( \displaystyle (0, 3, 3),\ (-3, 1, -1),\ (3, 0, -3) \).
Problem 9.44medium✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-2, 0, 3),\ (-2, 1, 1),\ (-2, -1, 2) \).
Problem 9.76mediumReviewed
Find an equation of the plane through the points \( \displaystyle (-1, -3, 1),\ (-1, 2, 3),\ (-1, 0, 3) \).
Problem 9.81medium✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (0, 2, -1),\ (3, -1, 0),\ (-1, 3, 1) \).
Problem 9.82mediumNot checked
Find an equation of the plane through the points \( \displaystyle (3, -3, -3),\ (-3, 3, 1),\ (2, -2, 3) \).
Problem 9.87mediumNot checked
Find an equation of the plane through the points \( \displaystyle (0, 1, 0),\ (-1, 3, -2),\ (0, 2, 0) \).
Problem 9.88mediumNot checked
Find an equation of the plane through the points \( \displaystyle (0, -3, -1),\ (0, 2, 0),\ (0, -1, -2) \).
Problem 9.110medium✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (1, -2, -1),\ (1, -2, -2),\ (0, 3, 1) \).
Problem 9.111mediumNot checked
Find an equation of the plane through the points \( \displaystyle (-1, -1, -2),\ (-2, 0, -2),\ (-1, -1, 1) \).
Problem 9.113mediumReviewed
Find an equation of the plane through the points \( \displaystyle (-1, 1, 3),\ (-1, 1, -2),\ (0, 2, 1) \).
Problem 9.114mediumNot checked
Find an equation of the plane through the points \( \displaystyle (2, -2, 3),\ (-2, 2, 3),\ (-1, -3, 1) \).
Problem 9.159mediumReviewed
Find an equation of the plane through the points \( \displaystyle (-1, -3, 2),\ (3, -3, -2),\ (-3, -1, 2) \).
Problem 9.160medium✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (3, 1, 3),\ (0, 3, -1),\ (-2, 3, -1) \).
Problem 9.179mediumReviewed
Find an equation of the plane through the points \( \displaystyle (1, 1, 1),\ (-3, 1, 1),\ (3, -1, 2) \).
Problem 9.204mediumNot checked
Find an equation of the plane through the points \( \displaystyle (2, 1, -2),\ (2, 1, -1),\ (-2, -2, -2) \).
Problem 9.207medium✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (3, -3, -3),\ (2, -3, 1),\ (3, -3, 1) \).
Problem 9.209mediumNot checked
Find an equation of the plane through the points \( \displaystyle (1, -2, 1),\ (-1, 1, 2),\ (-1, 3, 2) \).
Problem 9.34hardNot checked
Find an equation of the plane through the points \( \displaystyle (-2, 1, -1),\ (-3, 3, 1),\ (-1, 3, 3) \).
Problem 9.37hardNot checked
Find an equation of the plane through the points \( \displaystyle (-1, -2, 0),\ (-3, 3, 2),\ (0, 0, -3) \).
Problem 9.39hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-2, 0, 3),\ (-3, 3, -3),\ (1, 3, -3) \).
Problem 9.42hardReviewed
Find an equation of the plane through the points \( \displaystyle (2, 0, -1),\ (2, -3, 1),\ (-2, 3, 2) \).
Problem 9.45hardReviewed
Find an equation of the plane through the points \( \displaystyle (-2, -1, -2),\ (-3, 2, 2),\ (1, -1, 1) \).
Problem 9.80hardReviewed
Find an equation of the plane through the points \( \displaystyle (-2, 2, -3),\ (0, 2, -2),\ (-2, -1, 0) \).
Problem 9.83hardNot checked
Find an equation of the plane through the points \( \displaystyle (2, 0, -3),\ (3, 0, -1),\ (3, 2, 3) \).
Problem 9.107hardNot checked
Find an equation of the plane through the points \( \displaystyle (-3, 0, 3),\ (3, -2, -1),\ (-3, 1, -1) \).
Problem 9.131hardReviewed
Find an equation of the plane through the points \( \displaystyle (1, -1, 1),\ (-2, 2, -3),\ (-2, 3, 2) \).
Problem 9.134hardNot checked
Find an equation of the plane through the points \( \displaystyle (0, 2, 0),\ (2, 3, 2),\ (-1, 2, -2) \).
Problem 9.135hardReviewed
Find an equation of the plane through the points \( \displaystyle (1, 2, 2),\ (-2, 1, -3),\ (-1, -2, -3) \).
Problem 9.136hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (2, 1, -2),\ (3, 0, -3),\ (3, -1, -2) \).
Problem 9.137hardNot checked
Find an equation of the plane through the points \( \displaystyle (-1, 3, -3),\ (-2, -3, -2),\ (2, 1, -1) \).
Problem 9.138hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (3, 2, -2),\ (2, 2, 1),\ (2, 3, -3) \).
Problem 9.155hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-3, -2, 2),\ (2, 0, 0),\ (2, 3, -1) \).
Problem 9.156hardReviewed
Find an equation of the plane through the points \( \displaystyle (-1, -1, 0),\ (1, 3, -1),\ (-3, 2, 3) \).
Problem 9.157hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-2, 1, 2),\ (2, -1, 2),\ (-3, -1, 0) \).
Problem 9.158hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (2, 1, 0),\ (1, 1, 0),\ (0, 2, -2) \).
Problem 9.161hardNot checked
Find an equation of the plane through the points \( \displaystyle (2, -3, 2),\ (-1, 1, -3),\ (3, 1, 0) \).
Problem 9.162hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (3, -2, 3),\ (-2, -3, 1),\ (-1, -1, -3) \).
Problem 9.180hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (2, 0, -3),\ (1, -2, 1),\ (3, -2, 3) \).
Problem 9.181hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-2, 1, 3),\ (3, 3, -1),\ (-2, -3, 2) \).
Problem 9.182hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-3, 3, -3),\ (-1, -1, 1),\ (-2, -2, -3) \).
Problem 9.183hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-1, 2, -1),\ (-3, 3, 2),\ (2, 1, -1) \).
Problem 9.184hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (3, 0, 2),\ (-2, 1, -2),\ (-2, -2, 0) \).
Problem 9.185hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-3, -1, -1),\ (2, -1, -3),\ (-2, -3, 3) \).
Problem 9.186hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-2, -3, 1),\ (1, -1, 3),\ (3, 2, -1) \).
Problem 9.203hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (3, 1, -3),\ (-2, -1, 0),\ (3, -2, -1) \).
Problem 9.205hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (-3, 2, -2),\ (0, 3, -2),\ (-3, -2, 0) \).
Problem 9.206hard✓ Nihil obstat
Find an equation of the plane through the points \( \displaystyle (1, -3, 2),\ (-2, 2, -2),\ (0, 1, 3) \).
Problem 9.208hardReviewed
Find an equation of the plane through the points \( \displaystyle (3, 3, -3),\ (2, -3, 0),\ (3, 1, 3) \).
Problem 9.210hard✓ Nihil obstat