Let L be the line of intersection of the planes 2x+3y+z=1\; \; and\; \; x+3y+2z=2. If L makes an angle \alpha with the positive x-axis,then \cos \alpha equals

  • Option 1)

    1

  • Option 2)

    \frac{1}{\sqrt{2}}

  • Option 3)

    \frac{1}{\sqrt{3}}

  • Option 4)

    \frac{1}{2}

 

Answers (1)

As we learnt in 

Direction Cosines -

i)    \sin^{2} \alpha+ \sin^{2} \beta+\sin^{2} \gamma= 2

ii)    If OP =r then the co-ordinates of P will be (lr,mr,nr)

iii)    Direction cosines of X-axis are (1,0,0)

iv)    Direction cosines of Y-axis are (0,1,0)

v)    Direction cosines of Z-axis are (0,0,1)

-

 

 Let DCs of L be (l,m,n) then 2l+3m+2n=0 and l + 3m + 2n = 0 

On solving

\frac{l}{3}=\frac{m}{-3}=\frac{n}{3}

So,\: \: \: \: l:m:n= \frac{l}{\sqrt{}3},\frac{-1}{\sqrt3},\frac{1}{\sqrt3}

cos\alpha =\frac{1}{\sqrt{3}}


Option 1)

1

Incorrect Option

 

Option 2)

\frac{1}{\sqrt{2}}

Incorrect Option

 

Option 3)

\frac{1}{\sqrt{3}}

Correct Option

 

Option 4)

\frac{1}{2}

Incorrect Option

 

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