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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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DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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