If the linex-x1a1 - y-y1b1 - z-z1c1makes an angle - with the plane a2x - b2y - c2z - d- then which of the following is correct -

We know in the given equation of line a_1, b_1, c_1 nbsp;are the direction ratios of the line and in the equation of plane nbsp;a_2, b_2, c_2 are the nbsp ...

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Byju's Answer

Standard XIII

Mathematics

Angle between a Plane and a Line

If the linex-...

Question

If the line
$\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}$
makes an angle $θ$ with the plane $a_{2} x + b_{2} y + c_{2} z = d$ , then which of the following is correct -

cos (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}

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sin (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}

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tan (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}

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cot (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}

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Solution

The correct option is B $sin (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}$
We know in the given equation of line $a_{1}, b_{1}, c_{1}$ are the direction ratios of the line and in the equation of plane $a_{2}, b_{2}, c_{2}$ are the direction ratios of the normal to the plane. So we have two vectors with us. All we need now is the angle between them.
If the angle between plane and line is $θ$ , then the angle between normal to the plane and line is $90^{\circ} - θ$ .
$cos (90^{\circ} - θ) = \frac{a . b}{| a | | b |}$
$cos (90^{\circ} - θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}$

$\to sin (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}$

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Similar questions

Q. If the line

\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}

makes an angle

θ

with the plane

a_{2} x + b_{2} y + c_{2} z = d

, then which of the following is correct -

Q. If the line

\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}

makes an angle

θ

with the plane

a_{2} x + b_{2} y + c_{2} z = d

, then =

\frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}

Q. If a line

\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}

lies in a plane

a_{2} x + b_{2} y + c_{2} z = d,

then which of the following is / are correct -

Q. Equation/equations of the planes bisecting the angles between the planes

a_{1} x + b_{1} y + c_{1} z + d_{1} = 0

and

a_{2} x + b_{2} y + c_{2} z + d_{2} = 0

is given by -

Let a, b, c are the direction ratios of the normal and (x, y, z) be the co-ordinate of the point through which the plane passes. Then the equation of plane is:

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If the line x−x1a1=y−y1b1=z−z1c1 makes an angle θ with the plane a2x+b2y+c2z=d, then which of the following is correct -

If the line
$\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}$
makes an angle $θ$ with the plane $a_{2} x + b_{2} y + c_{2} z = d$ , then which of the following is correct -