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Vector Algebra

Question
CBSEENMA12034011
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Show that the points A (2, 6, 3). B (1, 2, 7) and C (3, 10, – 1) are collinear.

Solution
The given points are A (2, 6, 3), B (1, 2, 7) , C (3, 10, –1).
Let O be origin.
therefore space space space space OA with rightwards arrow on top space equals space 2 space straight i with hat on top plus space 6 space straight j with hat on top space plus space space 3 space straight k with hat on top comma space space space OB with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 7 space straight k with hat on top space space space space space space space space OC with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 10 space straight j with hat on top space minus space straight k with hat on top space space space space space space space AB with rightwards arrow on top space equals space OB with rightwards arrow on top space minus space OA with rightwards arrow on top space equals space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 7 space straight k with hat on top right parenthesis space minus space left parenthesis 2 space straight j with hat on top space plus space 6 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus straight i with hat on top space minus space 4 space straight j with hat on top space plus space 4 space straight k with hat on top space space space space space space space BC with rightwards arrow on top space equals space OC with rightwards arrow on top space minus space OB with rightwards arrow on top space equals space left parenthesis 3 straight i with hat on top space plus space 10 straight j with hat on top space minus space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 7 space straight k with hat on top right parenthesis space space space space space space space space space space space space space space space equals space 2 straight i with hat on top space plus space 8 space straight j with hat on top space minus space space 8 space straight k with hat on top therefore space space space space space space BC with rightwards arrow on top space equals space minus 2 space AB with rightwards arrow on top space space space space space space space space space space rightwards double arrow space space space space BC with rightwards arrow on top space equals space 2 space BA with rightwards arrow on top because space space space space space space BC with rightwards arrow on top space and space BA with rightwards arrow on top space are space parallel space vectors. space
But B is their common point
∴    points A, B, C are collinear.

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