Question 1:Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, – 1), (4, 3, – 1).
Question 2:If l 1 , m1 , n1 and l 2 , m2 , n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are 1 2 2 1 12 21 1 2 2 1 m n m n nl n
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Question 3:Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.
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Question 4:Find the equation of a line parallel to x-axis and passing through the origin.
Question 5:If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
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Question 6:If the lines 1 23 1 1 6 and 32 2 3 1 5 x y z xy z k k are perpendicular, find the value of k.
Question 7:Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane ) 9 0 ˆ r (. i ˆ 2 ˆj 5 k .
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Question 8:Find the equation of the plane passing through (a, b, c) and parallel to the plane ˆ ˆ ˆ ri jk ( ) 2.
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Question 9:Find the shortest distance between lines ˆˆ ˆ ˆ ˆ ˆ ri j k i j k 6 2 2 ( 2 2) and r i 4 ( ˆ ˆ k i jk ˆ ˆ 3 2 ˆ 2 ) .
Question 10:Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane.
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Question 11:Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.
Question 12:Find the coordinates of the point where the line through (3, – 4, – 5) and (2, – 3, 1) crosses the plane 2x + y + z = 7.
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Question 13:Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
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Question 14:If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane ˆ ˆ ˆ (3 4 12 ) 13 0, ri j k then find the value of p.
Question 15:Find the equation of the plane passing through the line of intersection of the planes ˆ ˆ ˆ ri jk ( ) 1 and ˆ ˆ ˆ r i jk (2 3 ) 4 0 and parallel to x-axis.
1. Show that the line joining the origin to the po
Question 16:If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of the plane passing through P and perpendicular to OP.
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Question 17:Find the equation of the plane which contains the line of intersection of the planes ˆ ˆ ˆ ri j k ( 2 3) 4 0 , ˆ ˆ ˆ r i jk (2 ) 5 0 and which is perpendicular to the plane ˆ ˆ ˆ ri jk (5 3 6 ) 8 0
Question 18:Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the line ˆˆ ˆ ˆ ˆ ˆ r ij k i j k 2 2( 3 4 2 ) and the plane ˆ ˆ ˆ ri jk ( ) 5 .
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Question 19:Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes ˆ ˆ ˆ ri j k ( 2 ) 5 and ˆ ˆ ˆ r i jk (3 ) 6 .
Question 20:Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines: 7 10 16 19 3 8 x y z and 15 3 x = 29 5 8 5 y z .
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Question 21:Prove that if a plane has the intercepts a, b, c and is at a distance of π units from the origin, then 2 2 2 2 1 1 1 1 a b c p . Choose the correct answer in Exercises 22 and 23
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Question 22:Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is (A) 2 units (B) 4 units (C) 8 units (D) 2 29 units
Question 23:The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through 5 0,0, 4
