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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.

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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 $AB with rightwards arrow on top space equals space stack straight A apostrophe space straight B apostrophe with rightwards arrow on top$

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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.

Here $PQ with rightwards arrow on top space equals space AP with rightwards arrow on top space plus space PB with rightwards arrow on top space plus space PC with rightwards arrow on top$ (given)
$therefore space space space PQ with rightwards arrow on top space minus space PC with rightwards arrow on top space equals space AP with rightwards arrow on top space plus space PB with rightwards arrow on top therefore space space PQ with rightwards arrow on top plus space CP with rightwards arrow on top space equals space AP with rightwards arrow on top space plus space PB with rightwards arrow on top therefore space space space space space CQ with rightwards arrow on top space equals space AB with rightwards arrow on top space space space space space space space space space space space space rightwards double arrow space space space space CQ space equals space AB space and space CQ space vertical line vertical line thin space AB$
∴ ABQC is a parallelogram
But A. B, C are given to be fixed points and ABQC is a || gm
∴ Q is a fixed point.