2011_Tutorial_6 - . (i) 2 x + 3 y + 9 z = 15; (ii) 15 / √...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 6 1. Write the vector 3 i +2 j + k as a sum of two vectors u + v such that u is parallel to w = i +3 j +4 k and v is perpendicular to w . (Hint: Use projection) Ans . 1 2 ( i + 3 j + 4 k ) + 1 2 (5 i + j - 2 k ) 2. Consider the two lines: 1 : x = 2 + 2 t, y = 2 + t, z = 3 + 3 t 2 : x = - 12 + 4 t, y = - 5 + 2 t, z = - 3 + t Show that 1 and 2 intersect. Find the point of intersection and an equation of the plane containing 1 and 2 . Ans . (0 , 1 , 0) , - x + 2 y = 2 3. (i) Find an equation of the plane Π passing through the points A (3 , 3 , 0) ,B (3 , 0 , 1) and C (0 , 2 , 1). (ii) Find the distance of Π from O (0 , 0 , 0). (iii) Let D = (4 , 2 , 1). Find the coordinates of the point of intersection of the plane Π in part (i) and the line segment OD . Ans
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Unformatted text preview: . (i) 2 x + 3 y + 9 z = 15; (ii) 15 / √ 94; (iii) 15 23 (4 , 2 , 1) 4. Find the shortest distance between the two planes: Π 1 : 2 x + 2 y-z = 1 and Π 2 : 4 x + 4 y-2 z = 5 . Ans . 1/2 5. Two particles travel along the space curves which, at time t , are given by: r 1 ( t ) = t i + t 2 j + t 3 k , r 2 ( t ) = (1 + 2 t ) i + (1 + 6 t ) j + (1 + 14 t ) k . Do the particles collide? Do their paths intersect? 6. A and B are two non-zero constant vectors and || B || = 1. If the angle between them is equal to π 4 , find the value of lim x → || A + x B ||-|| A || x Ans . √ 2 2...
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