Show that the points are collinear, where are three non-coplanar vectors. - Wired Faculty

Question

Show that the points $straight a with rightwards arrow on top space minus space 2 space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top comma space space space space 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space minus space 4 space straight c with rightwards arrow on top comma space minus 7 space straight b with rightwards arrow on top space plus space 10 space straight c with rightwards arrow on top$ are collinear, where $straight a with rightwards arrow on top comma space space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top$ are three non-coplanar vectors.

Solution

Let $straight a with rightwards arrow on top space minus space 2 straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top comma space space 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space minus space 4 space straight c with rightwards arrow on top comma space space minus 7 space straight b with rightwards arrow on top space plus space 10 space straight c with rightwards arrow on top$ be position vectors of A, B and C respectively.
Now,    $AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space open parentheses 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space minus space 4 space straight c with rightwards arrow on top close parentheses space minus space open parentheses straight a with rightwards arrow on top space minus space 2 space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top close parentheses$
   $therefore space space space AB with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space 5 space straight b with rightwards arrow on top space minus space 7 space straight c with rightwards arrow on top$
Now, $AC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight A$
   $equals space open parentheses negative 7 space straight b with rightwards arrow on top space plus space 10 space straight c with rightwards arrow on top close parentheses space minus space open parentheses straight a with rightwards arrow on top space minus space 2 space straight b with rightwards arrow on top space plus space 3 space straight c with rightwards arrow on top close parentheses space equals space minus straight a with rightwards arrow on top space minus space 5 space straight b with rightwards arrow on top space plus space 7 space straight c with rightwards arrow on top$
$therefore space space space space AC with rightwards arrow on top space equals space minus AB with rightwards arrow on top space space space space rightwards double arrow space space space space AB with rightwards arrow on top space and space space AC with rightwards arrow on top space are space parallel space vectors$
But these vectors have the same initial point
∴ point A, B and C are collinear.

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Vector Algebra

Show that the points are collinear, where are three non-coplanar vectors.

Some More Questions From Vector Algebra Chapter

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