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c. a. ~. ~. ( a + b ). ~ ~. Example 3 : solution. A. 2. E. 1. 1. 1- . C. D. . 1- . F. 1. . B. c. a. b. a. c. b. a. c. ~. ~. ~. ~. ~. ~. ~. ~. A. ( a + b ). 2. ~ ~. Since a , b and c are non-parallel vectors,. E. 1. 1. ~ ~ ~.

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  1. c a ~ ~ (a + b) ~ ~ Example 3 : solution A 2 E 1 1 1- C D  1-  F 1  B

  2. c a b a c b a c ~ ~ ~ ~ ~ ~ ~ ~ A (a + b) 2 ~ ~ Since a, b and care non-parallel vectors, E 1 1 ~ ~ ~ 1- C D  1-  F 1  B ~ ~ ~

  3. Example 35:The eqn of 3 planes p1, p2, p3 are 2x 5y + 3z = 3, 3x + 2y 5z = 5, 5x + y + 17z =  . When  =  20.9 and  = 16.6, find the pt at which planes meet. Soln When  =  20.9 and  = 16.6, We have eqns: 2x 5y + 3z = 3, 3x + 2y 5z = 5, 5x 20.9y + 17z = 16.6 Using GC Plysmlt2, x = - 4/11, y = - 4/11, z = 7/11 Hence, pt of intersection is

  4. Example 35 (con’t): The planes p1 and p2 intersect in a line l. (i) Find a vector eqn of l. Soln (i) p1: 2x 5y + 3z = 3, p2 : 3z + 2y 5z = 5 Using GC Plysmlt2,

  5. Example 35 (con’t):(ii) Given that all 3 planes meet in the line l, find and. Soln(ii) Given p3: 5x + y + 17z =  So l must lie on p3. So normal of p3 is  l, i.e.  5 +  + 17 = 0   = - 22   = 17

  6. Example 35 (con’t): (iii) Given instead that the 3 planes hv no point in common, what can be said about and? Soln (iii) If there is no pt in common. Since p1 and p2 intersect at l. Hence, l can be parallel to p3, so  can be - 22 But l should not lie on p3,so cannot be 17.

  7. Example 35 (con’t): (iv) Find the cartesian eqn of the plane which contains l and the point (1, 1, 3). Soln l Cartesian eqn: 3x – y – 2z = –2

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