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Three Dimensional Geometry

Short Answer Type

71. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

Let $stack straight m subscript 1 with rightwards arrow on top space space and space stack straight m subscript 2 with rightwards arrow on top$ be vectors parallel to the two given lines.
$therefore space space space space space space space stack straight m subscript 1 with rightwards arrow on top space equals space straight a space straight i with hat on top space plus space straight b space straight j with hat on top space plus space straight c space straight k with hat on top$
and $stack straight m subscript 2 with rightwards arrow on top space equals space left parenthesis straight b minus straight c right parenthesis space straight i with hat on top space plus space left parenthesis straight c minus straight a right parenthesis space straight j with hat on top space plus space left parenthesis straight a minus straight b right parenthesis space straight k with hat on top$
Let θ be angle between given lines.
∴ θ is angle between $stack straight m subscript 1 with rightwards arrow on top space space and space stack straight m subscript 2 with rightwards arrow on top.$
$therefore space space space cos space straight theta space equals space fraction numerator stack straight m subscript 1 with rightwards arrow on top. space stack straight m subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight m subscript 1 with rightwards arrow on top close vertical bar space open vertical bar stack straight m subscript 2 with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator straight a space left parenthesis straight b minus straight c right parenthesis space plus space straight b space left parenthesis straight c minus straight a right parenthesis space plus space straight c space left parenthesis straight a minus straight b right parenthesis over denominator square root of straight a squared plus straight b squared plus straight c squared end root space square root of left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared plus left parenthesis straight a minus straight b right parenthesis squared end root end fraction space space space space space space space space space space space space space space space space space space equals space fraction numerator ab minus ca plus bc minus ab plus ca minus ab over denominator square root of straight a squared plus straight b squared plus straight c squared end root space square root of left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared plus left parenthesis straight a minus straight b right parenthesis squared end root end fraction equals 0 therefore space space space space space straight theta space equals space straight pi over 2$