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
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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 are any four vectors in 3 - dimensional space with the same initial point and such that 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 respectively are such that . 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
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.
When P lies outside AB and is nearer to A than B. then AP<PB. Also division is external ∴ – 1 < λ < 0 When P is nearer to B than A, the AP > PB ∴ λ < – I
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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.