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Vector Algebra

Short Answer Type

111. Show that the three points A (1, –2, –8) , B (5. 0. –2) and C (11, 3. 7) are collinear and find the ratio in which B divides AC.

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112. If Q is the point of intersection of the medians of the triangle ABC, then prove that $QA with rightwards arrow on top space plus space QB with rightwards arrow on top space plus space QC with rightwards arrow on top space equals space 0 with rightwards arrow on top.$

Let $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top$ be the position vectors of vertices A, B, C respectively. Then position vector of centroid Q is $fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 3 end fraction.$

L.H.S. = $QA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space QC with rightwards arrow on top$
$equals space left parenthesis straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight Q right parenthesis space plus space left parenthesis straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight Q right parenthesis space plus space left parenthesis straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight Q right parenthesis$
$equals space open parentheses straight a with rightwards arrow on top space minus space fraction numerator straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 3 end fraction close parentheses space plus space open parentheses straight b with rightwards arrow on top space minus space fraction numerator straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top over denominator 3 end fraction close parentheses space plus space open parentheses straight c with rightwards arrow on top space minus space fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 3 end fraction close parentheses equals space open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top close parentheses space minus space 3 space open parentheses fraction numerator straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top over denominator 3 end fraction close parentheses space equals space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top right parenthesis space minus space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top space equals space straight R. straight H. straight S.$

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113. ABCD is a parallelogram. E, F are mid-points of BC, CD respectively. AE, AF meet the diaginal BD at Q, P respectively. Show that PQ trisects DB.

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114. If

straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space stack straight c comma with rightwards arrow on top straight d with rightwards arrow on top

are any four vectors in 3 - dimensional space with the same initial point and such that

3 space straight a with rightwards arrow on top space space minus space 2 space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space minus space 2 space straight d with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space

show that terminals, A, B, C, D of these vectors are coplanar. Find the point at which AC and BD meet. Find the ratio in which P divides AC and BD.

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115. Four points A, B, C, D with position vectors

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 comma space straight d with rightwards arrow on top

respectively are such that

3 straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space plus space space 2 straight c with rightwards arrow on top space minus space 4 space straight d with rightwards arrow on top space equals space 0 with rightwards arrow on top

. Show that the four points are coplanar. Also, find the position vector of the point of intersection of lines AC and BD.

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Long Answer Type

116. The mid-points of two opposite sides of a quadrilateral and the mid-points of the diagonals are the vertices of a parallelogram. Prove using vectors.

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Short Answer Type

117. A point P divides a line segment AB in the ratio A : 1. Give the values of A for which
P lies in between AB and
(i) nearer A than B (ii) nearer B than A

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118. A point P divides a line segment AB in the ratio A : 1. Give the values of A for which P lies outside AB and
(i) nearer A than B (ii) nearer B than A.

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Long Answer Type

119. Show that the lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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120. Show that a quadrilateral is a parallelogram if an only if diagonals bisect each other.

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