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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.

Given points are A (1, –2, –8), B (5, 0, –2), C (11, 3, 7).
Let O be origin.
$OA with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 8 space straight k with hat on top comma space space OB with rightwards arrow on top space equals space 5 space straight i with hat on top space minus space 2 space straight k with hat on top comma space space OC with rightwards arrow on top space equals space 11 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 7 space straight k with hat on top AB with rightwards arrow on top space equals space OB with rightwards arrow on top space minus space OA with rightwards arrow on top space equals space left parenthesis 5 space straight i with hat on top space minus space 2 space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space minus space 2 space straight j with hat on top space minus space 8 space straight k with hat on top right parenthesis space equals space 4 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top AC with rightwards arrow on top space equals space OC with rightwards arrow on top space minus space OA with rightwards arrow on top space equals space left parenthesis 11 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 7 space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space minus space 2 space straight j with hat on top space minus space 8 space straight k with hat on top right parenthesis space space space space space space space equals space 10 space straight i with hat on top space plus space 5 space straight j with hat on top space plus space 15 space straight k with hat on top$

$therefore space space space AC with rightwards arrow on top space equals space 5 over 2 left parenthesis 4 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis space equals space 5 over 2 AB with rightwards arrow on top therefore space space space AC with rightwards arrow on top space and space AB with rightwards arrow on top space are space parallel space vectors$

But A is their common point.
∴ points A. B. C are collinear and B divides AC in the ratio 2 : 3.

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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.

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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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