Let a¯,b¯ and c¯ be the unit vectors such that a¯ and b¯ are mutually perpendicular and c¯ is equally inclined to a¯ and b¯ at angle θ . If c¯$$=xa$$¯$$+yb$$¯$$+z(a$$¯×b¯$$)$$, then

Question

Let a¯,b¯ and c¯ be the unit vectors such that a¯ and b¯ are mutually perpendicular and c¯ is  equally inclined to a¯ and b¯ at angle θ . If c¯$$=xa$$¯$$+yb$$¯$$+z(a$$¯×b¯$$)$$, then

Infinity Learn · Accepted Answer

Given that   a¯$$=b$$¯$$=c$$¯$$=1$$ and,  a¯.b¯$$=0$$⇒a¯,b¯$$=90$$∘As   c¯$$=xa$$¯$$+y$$ b¯$$+z$$ a¯×b¯ ⇒a¯c¯$$cos$$θ$$=x+y0+z0$$       by  taking   dot product with a   ⇒$$cos$$θ$$=x$$  Similarly,  $$cos$$θ$$=yc$$¯$$=xa$$¯$$+y$$ b¯$$+z$$ a¯×b¯  Squaring on both sidesc¯$$2=x2a$$−$$2+y2b$$−$$2+z2a$$¯×b¯$$2+2xya$$¯.b¯$$+2yz$$ b¯.a¯×b¯$$+2xz$$ a¯.a¯×b¯$$1=x2+y2+z2+0+0+0$$ ⇒$$1=cos2$$θ$$+cos2$$θ$$+z2$$ ⇒$$1$$−$$2cos2$$θ$$=z2$$ ⇒$$1$$−$$2x2=z2$$∴$$z2=1$$−$$2x2$$

$Let \bar{a}, \bar{b} and \bar{c} be the unit vectors such that \bar{a} and \bar{b} are mutually perpendicular and \bar{c} is$ $equally inclined to \bar{a} and \bar{b} at angle θ . If \bar{c} = x \bar{a} + y \bar{b} + z (\bar{a} \times \bar{b}), then$

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Let a¯,b¯ and c¯ be the unit vectors such that a¯ and b¯ are mutually perpendicular and c¯ is equally inclined to a¯ and b¯ at angle θ . If c¯=xa¯+yb¯+z(a¯×b¯), then

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$Let \bar{a}, \bar{b} and \bar{c} be the unit vectors such that \bar{a} and \bar{b} are mutually perpendicular and \bar{c} is$ $equally inclined to \bar{a} and \bar{b} at angle θ . If \bar{c} = x \bar{a} + y \bar{b} + z (\bar{a} \times \bar{b}), then$