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Planes: distances and angles practice problems

Planes: through a point with a given normal, the distance from a point, and the angle between two planes. 60 problems with worked solutions; in 60 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 9.629

medium
Find the angle between the planes \( \displaystyle - 3 x - 3 y - 4 z = 5 \) and \( \displaystyle - 3 x - 3 y - 2 z = 5 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{\left(-4\right) \left(-2\right) - 3 \left(-3\right) - 3 \left(-3\right)}\right|}{\sqrt{\left(-2\right)^{2} + \left(-3\right)^{2} + \left(-3\right)^{2}} \sqrt{\left(-3\right)^{2} + \left(-3\right)^{2} + \left(-4\right)^{2}}} = \frac{13 \sqrt{187}}{187} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(13*sqrt(187)/187).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{13 \sqrt{187}}{187} \right)} \approx 18.07^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.630

medium
Find the angle between the planes \( \displaystyle 3 x - y + 4 z = 0 \) and \( \displaystyle - x + y + z = -1 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{3 \left(-1\right) - 1 \cdot 1 + 4 \cdot 1}\right|}{\sqrt{\left(-1\right)^{2} + 1^{2} + 1^{2}} \sqrt{\left(-1\right)^{2} + 3^{2} + 4^{2}}} = 0 \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. \[ \operatorname{acos}{\left(0 \right)} = \frac{\pi}{2} \]
    θ = arccos of that.✓ Proved
Answer \( \frac{\pi}{2} \approx 90.00^\circ \)

✓ Nihil obstat Lines: 2 proved, 1 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 9.631

medium
Find the angle between the planes \( \displaystyle 2 x - 2 y + 3 z = 2 \) and \( \displaystyle - 2 x + 3 y - 3 z = -1 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{3 \left(-3\right) - 2 \cdot 3 + 2 \left(-2\right)}\right|}{\sqrt{\left(-2\right)^{2} + 3^{2} + \left(-3\right)^{2}} \sqrt{2^{2} + \left(-2\right)^{2} + 3^{2}}} = \frac{19 \sqrt{374}}{374} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(19*sqrt(374)/374).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{19 \sqrt{374}}{374} \right)} \approx 10.75^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.633

medium
Find the angle between the planes \( \displaystyle 3 x + 4 y + 4 z = 4 \) and \( \displaystyle - x + 3 y - 2 z = -3 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{4 \left(-2\right) + 3 \left(-1\right) + 4 \cdot 3}\right|}{\sqrt{\left(-1\right)^{2} + \left(-2\right)^{2} + 3^{2}} \sqrt{3^{2} + 4^{2} + 4^{2}}} = \frac{\sqrt{574}}{574} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(sqrt(574)/574).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{\sqrt{574}}{574} \right)} \approx 87.61^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.635

medium
Find the angle between the planes \( \displaystyle 3 x - 3 y + 3 z = 3 \) and \( \displaystyle - x + 3 y + 2 z = -1 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{\left(-3\right) 3 + 3 \left(-1\right) + 3 \cdot 2}\right|}{\sqrt{\left(-1\right)^{2} + 2^{2} + 3^{2}} \sqrt{3^{2} + \left(-3\right)^{2} + 3^{2}}} = \frac{\sqrt{42}}{21} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(sqrt(42)/21).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{\sqrt{42}}{21} \right)} \approx 72.02^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.715

medium
Find the angle between the planes \( \displaystyle 4 x + 2 y + 4 z = -1 \) and \( \displaystyle - 3 x + 2 y + 2 z = -2 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{4 \left(-3\right) + 2 \cdot 2 + 4 \cdot 2}\right|}{\sqrt{2^{2} + 2^{2} + \left(-3\right)^{2}} \sqrt{2^{2} + 4^{2} + 4^{2}}} = 0 \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. \[ \operatorname{acos}{\left(0 \right)} = \frac{\pi}{2} \]
    θ = arccos of that.✓ Proved
Answer \( \frac{\pi}{2} \approx 90.00^\circ \)

✓ Nihil obstat Lines: 2 proved, 1 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 9.772

medium
Find the angle between the planes \( \displaystyle - 4 x - 3 y + 2 z = 1 \) and \( \displaystyle 2 x - y - 2 z = -3 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{\left(-4\right) 2 + 2 \left(-2\right) - 3 \left(-1\right)}\right|}{\sqrt{\left(-1\right)^{2} + 2^{2} + \left(-2\right)^{2}} \sqrt{2^{2} + \left(-3\right)^{2} + \left(-4\right)^{2}}} = \frac{3 \sqrt{29}}{29} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(3*sqrt(29)/29).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{3 \sqrt{29}}{29} \right)} \approx 56.15^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.773

medium
Find the angle between the planes \( \displaystyle - 4 x - y - 3 z = 3 \) and \( \displaystyle 2 x - y - 2 z = 5 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{\left(-4\right) 2 - 1 \left(-1\right) - 3 \left(-2\right)}\right|}{\sqrt{\left(-1\right)^{2} + \left(-3\right)^{2} + \left(-4\right)^{2}} \sqrt{\left(-1\right)^{2} + 2^{2} + \left(-2\right)^{2}}} = \frac{\sqrt{26}}{78} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(sqrt(26)/78).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{\sqrt{26}}{78} \right)} \approx 86.25^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.817

medium
Find the angle between the planes \( \displaystyle 4 x + 2 y + 4 z = 5 \) and \( \displaystyle - 3 x + y + 2 z = 2 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{4 \left(-3\right) + 2 \cdot 1 + 4 \cdot 2}\right|}{\sqrt{1^{2} + 2^{2} + \left(-3\right)^{2}} \sqrt{2^{2} + 4^{2} + 4^{2}}} = \frac{\sqrt{14}}{42} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(sqrt(14)/42).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{\sqrt{14}}{42} \right)} \approx 84.89^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.885

medium
Find the angle between the planes \( \displaystyle - x + y + 3 z = -4 \) and \( \displaystyle x - y + z = -5 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{\left(-1\right) 1 + 1 \left(-1\right) + 3 \cdot 1}\right|}{\sqrt{\left(-1\right)^{2} + 1^{2} + 3^{2}} \sqrt{1^{2} + \left(-1\right)^{2} + 1^{2}}} = \frac{\sqrt{33}}{33} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(sqrt(33)/33).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{\sqrt{33}}{33} \right)} \approx 79.98^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.888

medium
Find the angle between the planes \( \displaystyle 4 x - 2 y - 2 z = -5 \) and \( \displaystyle - x + 3 y - z = -4 \).
  1. The angle between planes is the (acute) angle between their normals.
    Reviewed
  2. \[ \frac{\left|{\left(-2\right) 3 + 4 \left(-1\right) - 2 \left(-1\right)}\right|}{\sqrt{\left(-1\right)^{2} + \left(-1\right)^{2} + 3^{2}} \sqrt{\left(-2\right)^{2} + \left(-2\right)^{2} + 4^{2}}} = \frac{2 \sqrt{66}}{33} \]
    cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
  3. θ = arccos of that: acos(2*sqrt(66)/33).
    Reviewed
Answer \( \operatorname{acos}{\left(\frac{2 \sqrt{66}}{33} \right)} \approx 60.50^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 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 9.628

easy
Find the distance from \( \displaystyle (0, -2, 4) \) to the plane \( \displaystyle - 3 x + 4 y + 2 z = 0 \).
  1. Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
    Reviewed
  2. \[ \frac{\left|{4 \left(-2\right) + 0 - 3 \cdot 0 + 2 \cdot 4}\right|}{\sqrt{2^{2} + \left(-3\right)^{2} + 4^{2}}} = 0 \]
    Substitute.✓ Proved
Answer \( 0 \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.711

easy
Find the distance from \( \displaystyle (-4, -4, 1) \) to the plane \( \displaystyle 4 x + 4 y + 2 z = -6 \).
  1. Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
    Reviewed
  2. \[ \frac{\left|{4 \left(-4\right) + 4 \left(-4\right) + 2 \cdot 1 + 6}\right|}{\sqrt{2^{2} + 4^{2} + 4^{2}}} = 4 \]
    Substitute.✓ Proved
Answer \( 4 \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.723

easy
Find the distance from \( \displaystyle (-4, -4, -1) \) to the plane \( \displaystyle 2 x + 2 y - 3 z = 5 \).
  1. Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
    Reviewed
  2. \[ \frac{\left|{2 \left(-4\right) + 2 \left(-4\right) - 5 - 3 \left(-1\right)}\right|}{\sqrt{2^{2} + 2^{2} + \left(-3\right)^{2}}} = \frac{18 \sqrt{17}}{17} \]
    Substitute.✓ Proved
Answer \( \frac{18 \sqrt{17}}{17} \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.768

easy
Find the distance from \( \displaystyle (-3, 0, -1) \) to the plane \( \displaystyle 2 x + 3 y + 2 z = -2 \).
  1. Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
    Reviewed
  2. \[ \frac{\left|{2 \left(-3\right) + 2 \left(-1\right) + 3 \cdot 0 + 2}\right|}{\sqrt{2^{2} + 2^{2} + 3^{2}}} = \frac{6 \sqrt{17}}{17} \]
    Substitute.✓ Proved
Answer \( \frac{6 \sqrt{17}}{17} \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.774

easy
Find an equation of the plane through \( \displaystyle (-1, 1, 1) \) with normal vector \( \displaystyle \left\langle 3, -1, 4 \right\rangle \).
  1. n · (r − P) = 0, i.e. ax + by + cz = n · P.
    Reviewed
  2. \[ 3 \left(-1\right) - 1 \cdot 1 + 4 \cdot 1 = 0 \]
    n · P.✓ Proved
Answer \( 3 x - y + 4 z = 0 \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.776

easy
Find the distance from \( \displaystyle (4, -2, 0) \) to the plane \( \displaystyle 4 x + 4 y + 2 z = 5 \).
  1. Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
    Reviewed
  2. \[ \frac{\left|{4 \left(-2\right) - 5 + 2 \cdot 0 + 4 \cdot 4}\right|}{\sqrt{2^{2} + 4^{2} + 4^{2}}} = \frac{1}{2} \]
    Substitute.✓ Proved
Answer \( \frac{1}{2} \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.813

easy
Find an equation of the plane through \( \displaystyle (-3, -3, 3) \) with normal vector \( \displaystyle \left\langle 4, 3, -2 \right\rangle \).
  1. n · (r − P) = 0, i.e. ax + by + cz = n · P.
    Reviewed
  2. \[ 4 \left(-3\right) + 3 \left(-3\right) - 2 \cdot 3 = -27 \]
    n · P.✓ Proved
Answer \( 4 x + 3 y - 2 z = -27 \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.815

easy
Find an equation of the plane through \( \displaystyle (-3, 2, -2) \) with normal vector \( \displaystyle \left\langle 3, 4, 1 \right\rangle \).
  1. n · (r − P) = 0, i.e. ax + by + cz = n · P.
    Reviewed
  2. \[ 3 \left(-3\right) + 1 \left(-2\right) + 4 \cdot 2 = -3 \]
    n · P.✓ Proved
Answer \( 3 x + 4 y + z = -3 \)

✓ Nihil obstat Lines: 1 proved, 1 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 9.880

easy
Find the distance from \( \displaystyle (1, -1, -3) \) to the plane \( \displaystyle - 3 x + 3 y + 4 z = 0 \).
  1. Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
    Reviewed
  2. \[ \frac{\left|{4 \left(-3\right) - 3 \cdot 1 + 3 \left(-1\right) + 0}\right|}{\sqrt{\left(-3\right)^{2} + 3^{2} + 4^{2}}} = \frac{9 \sqrt{34}}{17} \]
    Substitute.✓ Proved
Answer \( \frac{9 \sqrt{34}}{17} \)

✓ Nihil obstat Lines: 1 proved, 1 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 planes: distances and angles practice

Find an equation of the plane through \( \displaystyle (3, 3, 3) \) with normal vector \( \displaystyle \left\langle 4, 2, 2 \right\rangle \).
Problem 9.626easy✓ Every equation proved
Find the distance from \( \displaystyle (3, -2, 3) \) to the plane \( \displaystyle - 4 x + 2 y + z = -4 \).
Problem 9.627easy✓ Every equation proved
Find the distance from \( \displaystyle (0, -3, 4) \) to the plane \( \displaystyle 3 x + 3 y + 2 z = -1 \).
Problem 9.632easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (0, -1, -2) \) with normal vector \( \displaystyle \left\langle -3, 4, 2 \right\rangle \).
Problem 9.638easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (-3, 2, 2) \) with normal vector \( \displaystyle \left\langle 3, 3, -1 \right\rangle \).
Problem 9.639easy✓ Every equation proved
Find the distance from \( \displaystyle (2, -4, 1) \) to the plane \( \displaystyle x + 4 y + 3 z = 2 \).
Problem 9.640easy✓ Every equation proved
Find the distance from \( \displaystyle (3, -1, -2) \) to the plane \( \displaystyle - x + 3 y + 2 z = -1 \).
Problem 9.713easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (-3, -1, -1) \) with normal vector \( \displaystyle \left\langle -4, 3, -3 \right\rangle \).
Problem 9.716easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (-1, -3, 2) \) with normal vector \( \displaystyle \left\langle -1, 1, -1 \right\rangle \).
Problem 9.717easy✓ Every equation proved
Find the distance from \( \displaystyle (3, 4, 0) \) to the plane \( \displaystyle - 2 x + 3 y - 2 z = 4 \).
Problem 9.719easy✓ Every equation proved
Find the distance from \( \displaystyle (-3, 4, -4) \) to the plane \( \displaystyle 4 x + 2 y - 3 z = 4 \).
Problem 9.722easy✓ Every equation proved
Find the distance from \( \displaystyle (1, -1, 2) \) to the plane \( \displaystyle - 2 x + 4 y + z = -1 \).
Problem 9.724easy✓ Every equation proved
Find the distance from \( \displaystyle (-3, 3, 2) \) to the plane \( \displaystyle 4 x + 3 y + 3 z = -6 \).
Problem 9.725easy✓ Every equation proved
Find the distance from \( \displaystyle (0, 2, -1) \) to the plane \( \displaystyle 2 x + y - z = -1 \).
Problem 9.767easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (3, -1, -2) \) with normal vector \( \displaystyle \left\langle 1, 2, -2 \right\rangle \).
Problem 9.771easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (3, 2, -3) \) with normal vector \( \displaystyle \left\langle 1, 1, -3 \right\rangle \).
Problem 9.775easy✓ Every equation proved
Find the distance from \( \displaystyle (1, 2, 1) \) to the plane \( \displaystyle 3 x + 3 y + 3 z = -5 \).
Problem 9.810easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (3, -2, 1) \) with normal vector \( \displaystyle \left\langle 2, 4, 1 \right\rangle \).
Problem 9.812easy✓ Every equation proved
Find the distance from \( \displaystyle (1, -1, -3) \) to the plane \( \displaystyle 2 x + 2 y + 2 z = 6 \).
Problem 9.818easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (0, -3, 2) \) with normal vector \( \displaystyle \left\langle -1, -1, -4 \right\rangle \).
Problem 9.882easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (2, 3, 0) \) with normal vector \( \displaystyle \left\langle -1, 2, -3 \right\rangle \).
Problem 9.883easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (0, -1, 2) \) with normal vector \( \displaystyle \left\langle 3, 1, -4 \right\rangle \).
Problem 9.884easy✓ Nihil obstat
Find an equation of the plane through \( \displaystyle (-1, -2, -3) \) with normal vector \( \displaystyle \left\langle 1, -1, 3 \right\rangle \).
Problem 9.886easy✓ Every equation proved
Find an equation of the plane through \( \displaystyle (1, -2, -3) \) with normal vector \( \displaystyle \left\langle -1, 3, -4 \right\rangle \).
Problem 9.887easy✓ Nihil obstat
Find the distance from \( \displaystyle (2, 4, -3) \) to the plane \( \displaystyle - 2 x - 3 y - 4 z = -4 \).
Problem 9.889easy✓ Every equation proved
Find the angle between the planes \( \displaystyle - x - 2 y - 2 z = 5 \) and \( \displaystyle 3 x + y + 2 z = 3 \).
Problem 9.634medium✓ Every equation proved
Find the angle between the planes \( \displaystyle x - 2 y + z = 2 \) and \( \displaystyle x - y + z = -5 \).
Problem 9.636medium✓ Every equation proved
Find the angle between the planes \( \displaystyle - 4 x + 2 y + z = 2 \) and \( \displaystyle - 2 x - y + z = 5 \).
Problem 9.637medium✓ Every equation proved
Find the angle between the planes \( \displaystyle 4 x - y + 2 z = 5 \) and \( \displaystyle 3 x - y - 2 z = -2 \).
Problem 9.712medium✓ Every equation proved
Find the angle between the planes \( \displaystyle 4 x - 4 y - 2 z = -5 \) and \( \displaystyle x - 2 y - 3 z = -2 \).
Problem 9.714medium✓ Every equation proved
Find the angle between the planes \( \displaystyle 4 x - 3 y + 3 z = 3 \) and \( \displaystyle 2 x - 2 y - 3 z = 1 \).
Problem 9.718medium✓ Every equation proved
Find the angle between the planes \( \displaystyle 2 x + 4 y - z = 0 \) and \( \displaystyle - 2 x + y + 3 z = -1 \).
Problem 9.720medium✓ Every equation proved
Find the angle between the planes \( \displaystyle - 4 x + 2 y - 4 z = -5 \) and \( \displaystyle x - 2 y + 2 z = -1 \).
Problem 9.721medium✓ Every equation proved
Find the angle between the planes \( \displaystyle x + 4 y + 4 z = 0 \) and \( \displaystyle x + 3 y + 2 z = 0 \).
Problem 9.769medium✓ Every equation proved
Find the angle between the planes \( \displaystyle x + 4 y - z = -5 \) and \( \displaystyle - 3 x - y + 3 z = -2 \).
Problem 9.770medium✓ Every equation proved
Find the angle between the planes \( \displaystyle - 3 x + 3 y - z = 2 \) and \( \displaystyle - 2 x - 3 y + 2 z = 5 \).
Problem 9.809medium✓ Every equation proved
Find the angle between the planes \( \displaystyle - x + 4 y - 4 z = -2 \) and \( \displaystyle - 3 x - 3 y - 2 z = -2 \).
Problem 9.811medium✓ Every equation proved
Find the angle between the planes \( \displaystyle x + 2 y - 4 z = 1 \) and \( \displaystyle 2 x - y - 2 z = 1 \).
Problem 9.814medium✓ Every equation proved
Find the angle between the planes \( \displaystyle x + 4 y + 4 z = -3 \) and \( \displaystyle - 2 x - y - 2 z = 4 \).
Problem 9.816medium✓ Every equation proved
Find the angle between the planes \( \displaystyle - 2 x - y + 2 z = -4 \) and \( \displaystyle - 3 x + 3 y - 2 z = 1 \).
Problem 9.881medium✓ Every equation proved