If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to Also, find the angle which with . - Wired Faculty

Question

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If $straight a with rightwards arrow on top comma straight b with rightwards arrow on top comma space and space straight c with rightwards arrow on top$ are mutually perpendicular vectors of equal magnitudes, show that the vector $stack straight a space with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top$ is equally $straight a with rightwards arrow on top comma straight b with rightwards arrow on top comma space and space straight c with rightwards arrow on top$ inclined to Also, find the angle which $stack straight a space with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top$ with $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space or space straight c with rightwards arrow on top$ .

Solution

Since $straight a with rightwards arrow on top space comma straight b with rightwards arrow on top space and space straight c with rightwards arrow on top$ are mutually perpendicular vectors, we have
$stack straight a. with rightwards arrow on top straight b with rightwards arrow on top space equals stack straight b. with rightwards arrow on top straight c with rightwards arrow on top space equals space stack straight c. with rightwards arrow on top straight a with rightwards arrow on top space equals space 0$
It is given that
$vertical line straight a with rightwards arrow on top vertical line space equals vertical line straight b with rightwards arrow on top vertical line space equals vertical line straight c with rightwards arrow on top vertical line Let space vector space straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top space be space inclined space to space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top space at space angles space straight theta subscript 1 comma straight theta subscript 2 space and space straight theta subscript 3 space respectively. Then comma cos space straight theta subscript 1 space equals space fraction numerator open parentheses straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top close parentheses. straight a with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight a with rightwards arrow on top vertical line end fraction space equals space fraction numerator straight a with rightwards arrow on top. straight a with rightwards arrow on top space plus straight b with rightwards arrow on top. straight c with rightwards arrow on top space plus straight c with rightwards arrow on top. straight a with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight a with rightwards arrow on top vertical line end fraction space equals space fraction numerator vertical line straight a with rightwards arrow on top vertical line squared over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight a with rightwards arrow on top vertical line end fraction space left square bracket straight b with rightwards arrow on top. straight a with rightwards arrow on top space equals space straight c with rightwards arrow on top. straight a with rightwards arrow on top space equals space 0 right square bracket equals fraction numerator vertical line straight a with rightwards arrow on top vertical line over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line end fraction cos space straight theta subscript 2 space equals space fraction numerator open parentheses straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top close parentheses. straight b with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line end fraction space equals space fraction numerator straight a with rightwards arrow on top. straight b with rightwards arrow on top space plus straight b with rightwards arrow on top. straight b with rightwards arrow on top space plus straight c with rightwards arrow on top. straight b with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line end fraction fraction numerator vertical line straight b with rightwards arrow on top vertical line squared over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line end fraction space left square bracket straight a with rightwards arrow on top. straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top. straight b with rightwards arrow on top space equals space 0 right square bracket equals fraction numerator vertical line straight b with rightwards arrow on top vertical line over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line end fraction cos space straight theta subscript 3 space equals space fraction numerator open parentheses straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top close parentheses. straight c with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line end fraction space equals space fraction numerator straight a with rightwards arrow on top. straight c with rightwards arrow on top space plus straight b with rightwards arrow on top. straight c with rightwards arrow on top space plus straight c with rightwards arrow on top. straight c with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line end fraction fraction numerator vertical line straight c with rightwards arrow on top vertical line squared over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line end fraction space left square bracket straight a with rightwards arrow on top. straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top. straight c with rightwards arrow on top space equals space 0 right square bracket equals fraction numerator vertical line straight c with rightwards arrow on top vertical line over denominator vertical line straight a with rightwards arrow on top space plus straight b with rightwards arrow on top space plus straight c with rightwards arrow on top vertical line end fraction Now comma space as space vertical line straight a with rightwards arrow on top vertical line space equals vertical line straight b with rightwards arrow on top vertical line space equals space vertical line straight c with rightwards arrow on top vertical line space comma cos space straight theta subscript 1 space equals space cosθ subscript 2 space equals space cos space straight theta subscript 3 therefore space straight theta subscript 1 space equals straight theta subscript 2 space equals straight theta subscript 3$

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