43. If the vector a = (x, y, z) mak...
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43. If the vector a = (x, y, z) makes equal angle with the vector b = (1, - 2, 3) and c = (2,3,- 1) and is perpendicular to the vector d = (1,-1,1) and then 124 cos? O is equal to

JEE/Engineering Exams
Maths
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( cos theta=frac{x(1)+y(-2)+z(3)}{sqrt{x^{2}+y^{2}+z^{2}} cdot sqrt{1^{2}+(-2)^{2}+(y)^{2}}}=frac{x-2 y+3 z}{sqrt{x^{2}+y^{2}+z^{2}} sqrt{14}} ) ( left.cos theta=frac{x(2)+y(3)+z(-1)}{sqrt{x^{2}+y^{2}+z^{2}} sqrt{14}}=frac{2 x+3 y-z}{sqrt{x^{2}+y^{2}+z} cdot sqrt{14}}right} ) ( cos 90=0 Rightarrow x(1) a+y(-1)+z(1)=0 ) ( 00 x+z=y ) ( 00 x-2 y+3 z=2 x+3 y-z ) ( 00 quad 4 z=x+5 y ) ( 4 z=y-z+5 y ) ( 0,5 z=6 y Rightarrow 00 z=frac{6 y}{5} cdot 4 x=-frac{y}{5} mid=7 )
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