Revision - Vectors & 3D Geometry

Grade 12 Maths Higher — Revision - Vectors & 3D Geometry: 25 practice questions with instant scoring and explanations.

1. If a⃗ = î + 2ĵ + 3k̂, then |a⃗| equals:
2. The unit vector in the direction of 3î + 4ĵ is:
3. If a⃗·b⃗ = 0 and a⃗, b⃗ are non-zero, then a⃗ and b⃗ are:
4. The value of î·(ĵ × k̂) is:
5. If a⃗ × b⃗ = 0 and a⃗, b⃗ are non-zero, then a⃗ and b⃗ are:
6. The projection of a⃗ = 2î + 3ĵ + 2k̂ on b⃗ = î + 2ĵ + k̂ is:
7. The angle between vectors a⃗ = î + ĵ and b⃗ = ĵ + k̂ is:
8. The area of the parallelogram with adjacent sides a⃗ = î + ĵ + k̂ and b⃗ = î - ĵ + k̂ is:
9. The volume of the parallelepiped with edges a⃗ = î + ĵ, b⃗ = ĵ + k̂, c⃗ = k̂ + î is:
10. The direction cosines of the vector 2î - 3ĵ + 6k̂ are:
11. The distance between the parallel planes 2x - 2y + z = 1 and 2x - 2y + z = 5 is:
12. The equation of the line through (1, 2, 3) parallel to b⃗ = î + 2ĵ - k̂ is:
13. The Cartesian equation of the plane passing through (1, 0, 0) with normal î + ĵ + k̂ is:
14. The angle between the planes 2x + y - z = 3 and x - y + 2z = 1 is:
15. The shortest distance between skew lines with direction vectors b⃗₁, b⃗₂ and position vectors a⃗₁, a⃗₂ is:
16. If a⃗, b⃗, c⃗ are mutually perpendicular unit vectors, then |a⃗ + b⃗ + c⃗| equals:
17. The position vector of the midpoint of the segment joining A(2, 3, -1) and B(-2, 1, 3) is:
18. The scalar triple product [a⃗ b⃗ c⃗] is zero when:
19. If a⃗ = 2î + 3ĵ, b⃗ = -î + 2ĵ, then a⃗ · b⃗ equals:
20. The equation of the plane containing the z-axis and making angle 60° with the plane x + y = 3 is:
21. The foot of perpendicular from (1, 2, 3) to the line x = y = z is:
22. The direction ratios of the line of intersection of planes x + y + z = 1 and x - y + z = 2 are:
23. The vector equation of the plane passing through the origin with normal 2î - 3ĵ + k̂ is:
24. If |a⃗| = 3, |b⃗| = 4 and |a⃗ × b⃗| = 6, then the angle between a⃗ and b⃗ is:
25. The line through (1, -1, 2) perpendicular to the plane 2x - 3y + 4z = 10 has direction ratios: