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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.

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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.

Take A as origin. Let

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

be the position vectors of B, C, D respectively. Let M be the point of intersection of diagonals of AC and BD.
In ∆ADC, P is the centroid as it is the point of intersection of two medians AF and DM as M is midpoint of AC.

therefore space space space AP with rightwards arrow on top space equals space fraction numerator 0 with rightwards arrow on top space plus space straight d with rightwards arrow on top space plus space straight c with rightwards arrow on top over denominator 3 end fraction space equals space fraction numerator 2 space straight d with rightwards arrow on top space plus straight b with rightwards arrow on top over denominator 3 end fraction

∴ P divides BD in the ratio 2 : 1
Similarly Q divides DB in the ratio 2 : 1
Hence the result.

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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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