Question

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
4x + 8y + z – 8 = 0 and y + z – 4 = 0

Solution

The equations of given planes are
4x + 8 y + z – 8 = 0    ...(1)
and 0x + y + z – 4 = 0    ...(2)
Direction ratios of normal to plane (1) are 4, 8, 1.
Direction ratios of normal to plane (2) are 0, 1, 1.
Now, $4 over 0 not equal to 8 over 1 not equal to 1 over 1$
∴ given planes (1) and (2) are not parallel.
Again (4) (0) + (8) (1) + (1) (1) = 0 + 8 + 1 = 9 ≠ 0
∴ given planes (1) and (2) are not perpendicular to each other.
Let θ be angle between planes.
$therefore space space space space cos space straight theta space equals space fraction numerator left parenthesis 4 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis 8 right parenthesis thin space left parenthesis 1 right parenthesis thin space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of left parenthesis 4 right parenthesis squared plus left parenthesis 8 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space square root of left parenthesis 0 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root end fraction$
   $equals space fraction numerator 0 plus 8 plus 1 over denominator square root of 16 plus 64 plus 1 end root space square root of 0 plus 1 plus 1 end root end fraction space equals fraction numerator 9 over denominator square root of 81 space square root of 2 end fraction space equals fraction numerator 9 over denominator 9 square root of 2 end fraction space equals fraction numerator 1 over denominator square root of 2 end fraction$
$therefore space space space straight theta space equals space 45 space degree.$

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Three Dimensional Geometry

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
4x + 8y + z – 8 = 0 and y + z – 4 = 0

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:4x + 8y + z – 8 = 0 and y + z – 4 = 0

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
4x + 8y + z – 8 = 0 and y + z – 4 = 0