If A and B are two points in space and if they become respectiv
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    Vector Algebra

    Multiple Choice QuestionsShort 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.
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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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    Multiple Choice QuestionsLong 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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    Multiple Choice QuestionsShort 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

    After translation, the positions of A, B are . A', B’ respectively.
    therefore space space space space AA apostrophe space equals space BB apostrophe space and space AA apostrophe space vertical line vertical line thin space BB apostrophe therefore space space space ABB apostrophe straight A apostrophe space is space straight a space space parllelogram therefore space space space AB space equals space straight A apostrophe straight B apostrophe space and space AB vertical line vertical line straight A apostrophe straight B apostrophe rightwards double arrow space space space space AB with rightwards arrow on top space equals stack straight A apostrophe space straight B apostrophe with rightwards arrow on top.

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    40. ABC is a triangle and P any point in BC. If PQ with rightwards arrow on top is the sum of AP with rightwards arrow on top comma space PB with rightwards arrow on top comma space PC with rightwards arrow on top comma show that ABQC is a || gm and Q. therefore a fixed point.
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