If the line x-x1-a1-y-y1-b1-z-z1-c1 makes an angle - with the plane a2 x-b2 y-c2 z-d- then -a1 a2-b1 b2-c1 c2-a12-b12-c12a22-b22-c22

The correct option is A sin θIf the angle between plane and line is θ, then the angle nbsp;between normal to the plane and line is 90^∘ θ. cos(90^∘ θ) = ...

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Byju's Answer

Standard XII

Mathematics

Angle between a Plane and a Line

If the line x...

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

s i n θ

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c o s θ

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t a n θ

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s e c θ

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Solution

The correct option is A $s i n θ$
If the angle between plane and line is $θ$ , then the angle between normal to the plane and line is $90^{\circ} - θ$ .

$cos (90^{\circ} - θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}$
Or $sin (θ) = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})}}$

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If the line x−x1a1=y−y1b1=z−z1c1 makes an angle θ with the plane a2x+b2y+c2z=d, then = a1a2+b1b2+c1c2√(a21+b21+c21)(a22+b22+c22)

The correct option is A sin θIf the angle between plane and line is θ, then the angle between normal to the plane and line is 90∘−θ. cos(90∘−θ)=a1a2+b1b2+c1c2√(a21+b21+c21)(a22+b22+c22) Or sin(θ)=a1a2+b1b2+c1c2√(a21+b21+c21)(a22+b22+c22)

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