Vectors and Geometry
What this domain tests
Vectors in two and three dimensions: addition, scalar multiples, position and unit vectors, magnitude; the scalar product for angles and perpendicularity; the vector product for areas and normals; equations of lines and planes, plus angles and intersections between them.
Core rules
- a·b = |a||b|cos θ, so a·b = 0 exactly when a and b are perpendicular.
- a×b is perpendicular to both, with |a×b| = |a||b|sin θ equal to the parallelogram area; order matters, since b×a = -(a×b).
- Line: r = a + λd. Plane: r·n = d or r = a + λb + μc.
- Angle between a line and a plane: sin θ = |d·n|/(|d||n|), the complement of the angle with the normal.
- Distance from point P to plane r·n = d is |(p - a)·n|/|n|.
Worked example
Find the angle between a = i + 2j - 2k and b = 2i - j + k. a·b = (1)(2) + (2)(-1) + (-2)(1) = 2 - 2 - 2 = -2. |a| = √(1 + 4 + 4) = 3 and |b| = √(4 + 1 + 1) = √6. cos θ = -2/(3√6) = -0.2722, so θ = 105.8°, about 1.847 rad. The negative cosine correctly signals an obtuse angle.
Traps
- Using the cross product for angles or the dot product for areas.
- Reversing the order in a×b and dropping the direction sign.
- Omitting the modulus in |d·n| for a line-plane angle.
- Mixing up the position vector and the direction vector in r = a + λd.