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Example 23 - Find angle between 3x - 6y + 2z = 7 and 2x + 2y - Angle between two planes

  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise
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Example 23 Find the angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z =5. Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos θ = 𝑨﷮𝟏﷯ 𝑨﷮𝟐﷯ + 𝑩﷮𝟏﷯ 𝑩﷮𝟐﷯ + 𝑪﷮𝟏﷯ 𝑪﷮𝟐﷯﷮ ﷮ 𝑨﷮𝟏﷯﷮𝟐﷯ + 𝑩﷮𝟏﷯﷮𝟐﷯ + 𝑪﷮𝟏﷯﷮𝟐﷯﷯ ﷮ 𝑨﷮𝟐﷯﷮𝟐﷯ + 𝑩﷮𝟐﷯﷮𝟐﷯ + 𝑪﷮𝟐﷯﷮𝟐﷯﷯﷯﷯ Given the two planes are So, cos θ = 3 × 2﷯ + −6 × 2﷯ + (2 × −2)﷮ ﷮ 3﷮2﷯ + (−6)﷮2﷯ + 2﷮2﷯﷯ ﷮ 2﷮2﷯ + 2﷮2﷯ + (−2)﷮2﷯﷯﷯﷯ = 6 + −12﷯ + (−4)﷮ ﷮9 + 36 + 4﷯ × ﷮4 + 4 + 4﷯﷯﷯ = −10﷮ ﷮49 ﷯ × ﷮12﷯﷯﷯ = −10﷮7 × ﷮4×3﷯﷯﷯ = 10﷮7 × 2 × ﷮3﷯﷯ = 5﷮7 ﷮3﷯﷯ = 5﷮7 ﷮3﷯﷯ × ﷮3﷯﷮ ﷮3﷯﷯ = 5 ﷮3﷯﷮21﷯ So, cos θ = 5 ﷮3﷯﷮21﷯ ∴ θ = 𝒄𝒐𝒔﷮−𝟏﷯ 𝟓 ﷮𝟑﷯﷮𝟐𝟏﷯﷯ Therefore, the angle between the two planes is 𝑐𝑜𝑠﷮−1﷯ 5 ﷮3﷯﷮21﷯﷯

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