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HL Unit 8 (Vectors)
Figure 8.3: HL Spec ’00 Paper 2
(a) Find the coordinates of P , Q, R, and S. [Ans: (4, 0, −3), (3, 3, 0), (3, 1, 1), (5, 2, −1)] (b) Find an equation of the plane OAP B.
[Ans: 3x + 2y + 4z = 0]
(c) Calculate the volume, V , of the parallelepiped given that
−→ −−→ −−→
V = OA × OB · OC
[Ans: 15] 8.C-3 (HL 5/99) The coordinates of the points P , Q, R, and S are (4, 1, −1), (3, 3, 5), (1, 0, 2), and (1, 1, 2), respectively. (a) Find an equation of the line ` which passes through the point Q and −→ is parallel to the vector P R. (b) Find a cartesian equation of the plane π which contains the line ` and passes through the point S. [Ans: r = 3 (1 − t) i + (3 − t) j + (5 + 3t) k; 9x − 15y + 4z = 2] 8.C-4 (HL 5/03) The point A is the foot of the perpendicular from the point (1, 1, 9) to the plane 2x + y − z = 6. (a) Find n, the normal to the plane.
[Ans: h2, 1, −1i]
(b) Let p be the position vector for (1, 1, 9). Write an equation in terms of λ for a, where a is p plus an unknown multiple (λ) of a vector perpendicular to the plane. [Ans: (1 + 2λ) i + (1 + λ) j + (9 − λ) k] (c) Use the fact that A is in the plane to solve for λ, and then find the coordinates of A. [Ans: 2; (5, 3, 7)] (d) Check by showing that A satisfies the equation of the plane and that the vector from (1, 1, 9) to A is parallel to the normal of the plane. z y+1 = and the plane r · (i + 2j − k) = 1 2 3 intersects at the point P . Find the coordinates of P . [Ans: (2, 3, 1)]
8.C-5 (HL 5/04) The line x − 1 =
5−z x−3 = y+1 = and the plane 2x − y + 3z = 10 2 3 intersect at the point P . Find the coordinates of P . [Ans: p7, 1, −1]
8.C-6 (HL 5/05) The line
Mr. Budd, compiled September 29, 2010