Let [] be the angle between the lines whose direction cosines satisfy the equations l + m …

Question

Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations l + m - n = 0 and l2 + m2 - n2 = 0. Then the value of sin4$$\alpha$$ + cos4$$\alpha$$ is :

Accepted Answer

${l^2} + {m^2} + {n^2} = 1$ $$ \therefore $$ $2{n^2} = 1$ ($$ \because $$ l2 + m2 - n2 = 0) $$\Rightarrow n = \pm {1 \over {\sqrt 2 }}$$ $$ \therefore $$ $${l^2} + {m^2} = {1 \over 2}$$ \& $$l + m = {1 \over {\sqrt 2 }}$$ $$ \Rightarrow {1 \over 2} - 2lm = {1 \over 2}$$ $$ \Rightarrow lm = 0$$ or $m = 0$ $$ \therefore $$ $$l = 0,m = {1 \over {\sqrt 2 }}$$ or $$l = {1 \over {\sqrt 2 }}$$ $$ < 0,{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }} > $$ or $$ < {1 \over {\sqrt 2 }},0,{1 \over {\sqrt 2 }} > $$ $$ \therefore $$ $$\cos \alpha = 0 + 0 + {1 \over 2} = {1 \over 2}$$ $$ \therefore $$ $${\sin ^4}\alpha + {\cos ^4}\alpha = 1 - {1 \over 2}{\sin ^2}(2\alpha ) = 1 - {1 \over 2}.{3 \over 4} = {5 \over 8}$$

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