Maths · Vectors and 3-D geometry · chapter 10 of 13
Class 12 Maths Chapter 10: Vector Algebra case study question with answers
One case-based question of the kind the board asks: a passage, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.
Case study · 4 marks
A survey drone at a solar farm near Jodhpur lifts off from its charging pad, taken as the origin O. The x-axis points east, the y-axis north and the z-axis vertically up, with distances in metres. The drone first climbs along a straight line to A(6, 2, 3), where it photographs a row of panels, and then flies straight to B(6, 10, 9) to inspect an inverter shed. A relay beacon is to be placed at the mid-point C of AB.
Part 1 · 1 mark
The distance of A from the pad, in metres, is
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Answer: (b) 7
|→OA| = √(36 + 4 + 9) = √49 = 7 m.
Part 2 · 1 mark
The cosine of the angle that →OA makes with the vertical (the positive z-axis) is
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Answer: (c) 37
n = z|→OA| = 37.
Part 3 · 2 marks
Find the position vector of the beacon C and its distance from the pad.
Answer and marking scheme
→OC = 12(→OA + →OB) = 12(12î + 12ĵ + 12k̂) = 6î + 6ĵ + 6k̂. |→OC| = √108 = 6√3 m.
- mid-point formula gives →OC = 6î + 6ĵ + 6k̂ (1)
- |→OC| = 6√3 m (1)
Or
Find the area of the triangle OAB swept out between the pad and the two flight legs.
Answer and marking scheme
→AB = 8ĵ + 6k̂. →OA × →AB = |î ĵ k̂; 6 2 3; 0 8 6| = −12î − 36ĵ + 48k̂, of magnitude 12√26. Area = 12(12√26) = 6√26 m2.
- →OA × →AB = −12î − 36ĵ + 48k̂ (1)
- area 6√26 m^{2} (1)
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