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Vector Algebra

Long Answer Type

41. ABCD is a quadrilateral and O is any point in its plane. Show that it $OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space 0 with rightwards arrow on top$ then O is the point of intersection of lines joining the middle points of the opposite sides of ABCD.

Let E. F, G, H be mid-points of the sides AB. BC. CD. DA respectively of quadrilateral ABCD.
∴ EFGH is a || gm [From Co-ordinate Geometry]
∴ diagonals EG and FH bisect each other at P (say)
Now, $OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space plus space OD with rightwards arrow on top space equals space 0 with rightwards arrow on top$
$rightwards double arrow space space space open parentheses OA with rightwards arrow on top space plus space OB with rightwards arrow on top close parentheses space plus space open parentheses OC with rightwards arrow on top space plus space OD with rightwards arrow on top close parentheses space equals 0 with rightwards arrow on top rightwards double arrow space space space space 2. space OE with rightwards arrow on top space plus space space 2 space OG with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space space rightwards double arrow space space space space OE with rightwards arrow on top space plus space OG with rightwards arrow on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space space space 2 space OP with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight P space is space the space mid minus point space of space EG close square brackets rightwards double arrow space space space space space space space OP with rightwards arrow on top space equals space 0 with rightwards arrow on top$

∴ O coincides with P which is the point of intersection of lines joining the middle points of the opposite sides of ABCD.

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42. For any two vectors

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

, prove that
(i)

open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

(ii)

open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar

(iii)

open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space space space greater-than or slanted equal to space space open vertical bar straight a with rightwards arrow on top close vertical bar space minus space open vertical bar straight b with rightwards arrow on top close vertical bar

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43.

If $straight c with rightwards arrow on top space equals space 3 straight a with rightwards arrow on top space plus space 4 straight b with rightwards arrow on top$ and $2 straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top space minus space 3 straight b with rightwards arrow on top$ , show that
(i) $straight c with rightwards arrow on top space and space straight a with rightwards arrow on top$ have the same direction and $open vertical bar straight c with rightwards arrow on top close vertical bar space greater than space open vertical bar straight a with rightwards arrow on top close vertical bar$
(ii) $straight c with rightwards arrow on top space and space straight b with rightwards arrow on top$ have opposite direction and

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Short Answer Type

44. Show that the line joining the middle points of the consecutive sides of a quadrilateral is a parallelogram.

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Long Answer Type

45. In the figure, M is the mid-point of AB and N is the mid-point of CD and O is the mid-point of MN. Prove that
(i)

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 equals space straight O with rightwards arrow on top

(ii)

BC with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space 2 space MN with rightwards arrow on top

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46. Prove by vector method that the line segment joining the mid-points of the diagonals of trapezium is parallel to the parallel sides and equal to help of there difference.

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