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Vector Algebra

Short Answer Type

31. Prove that the line joining the middle points of two sides of a triangle is parallel to the third side and is half of it.

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32. ABC is a triangle and D is the mid-point of the side BC. Show that $AB with rightwards arrow on top space plus space AC with rightwards arrow on top space equals space 2 space AD with rightwards arrow on top.$

In $increment ABD comma space space space space space space space space space space space space space space space space space space AB with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space BD with rightwards arrow on top$ ...(1)
In $increment space ADC comma space space space space space space AC with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space DC with rightwards arrow on top$ ...(2)
Adding (1) and (2), we get

$AB with rightwards arrow on top space plus space AC with rightwards arrow on top space equals space 2 space AD with rightwards arrow on top space plus space open parentheses DB with rightwards arrow on top space plus space DC with rightwards arrow on top close parentheses$
$equals space 2 space AD with rightwards arrow on top space plus space DB with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight D space is space mid minus point space of space BC comma space space space space space because space space BD with rightwards arrow on top space equals DC with rightwards arrow on top close square brackets equals space 2 space stack AD space with rightwards arrow on top space minus space BD with rightwards arrow on top space plus space BD with rightwards arrow on top$

$therefore$ $AB with rightwards arrow on top space plus space AC with rightwards arrow on top space equals 2 space space AD with rightwards arrow on top$

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33. If D and E are the mid-points of sides AB and AC of a triangle ABC respectively show that

BE with rightwards arrow on top space plus space DC with rightwards arrow on top space equals space 3 over 2 BC with rightwards arrow on top.

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34.

If $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space and space straight d with rightwards arrow on top$ are distinct non-zero vectors represented by directed lines from the origin to the points A, B, C and D respectively and if $straight b with rightwards arrow on top minus straight a with rightwards arrow on top space equals space straight c with rightwards arrow on top space minus space straight d with rightwards arrow on top$ , then prove that ABCD is a parallelogram.

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35. Let O be the centre of regular hexagon ABCDEF. Find the sum.

OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space plus space stack OE space with rightwards arrow on top space plus space OF with rightwards arrow on top

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36. Show that the diagonals a quadrilateral bisect each other if and only if it is a parallelogram.

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Long Answer Type

37. 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.

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Short Answer Type

38. Show that the points

negative 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space plus space 5 space straight c with rightwards arrow on top comma space space straight a with rightwards arrow on top space plus 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 7 space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top

are collinear, where

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

are three non-coplanar vectors.

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39.

If A and B are two points in space and if they become respectively A', B' on a translation, show that $AB with rightwards arrow on top space equals space stack straight A apostrophe space straight B apostrophe with rightwards arrow on top$