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.
Since, ∴ P.V. of point dividing AC in the ratio 1 : 3 is the same as the P.V. of mid-point of BD. ∴ AC and BD intersect at P whose P.V. is This point P divides AC in the ratio 3:1 and BD in the ratio 1:1.
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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
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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.