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

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

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Example 35: The eqn of 3 planes p 1, p 2, p 3 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

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Example 35 (cont): The planes p 1 and p 2 intersect in a line l. (i) Find a vector eqn of l. Soln (i) p 1 : 2x 5y + 3z = 3, p 2 : 3z + 2y 5z = 5 Using GC Plysmlt2,

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Example 35 (cont):(ii) Given that all 3 planes meet in the line l, find and. Soln(ii) Given p 3 : 5x + y + 17z = So l must lie on p 3. So normal of p 3 is l, i.e. 5 + + 17 = 0 = - 22 = 17

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Example 35 (cont): (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 p 1 and p 2 intersect at l. Hence, l can be parallel to p 3, so can be - 22 But l should not lie on p 3, so cannot be 17.

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Example 35 (cont): (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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X = 2 + t y = -3 + 2t t = x – 2 t = (y + 3)/2 x – 2 = y + 3 2 2x – 4 = y + 3 y – 2x + 7 = 0 Finding the Cartesian Equation from a vector equation x = 2.

X = 2 + t y = -3 + 2t t = x – 2 t = (y + 3)/2 x – 2 = y + 3 2 2x – 4 = y + 3 y – 2x + 7 = 0 Finding the Cartesian Equation from a vector equation x = 2.

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