Vectors and Three-Dimensional Geometry
Vectors and Three-Dimensional Geometry tests vector algebra, scalar and vector products, direction ratios and cosines, equations of lines and planes, angles and distances. Keep vector notation consistent and distinguish a point from a direction vector.
Core rules
- If a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k, then a·b = a1b1+a2b2+a3b3 and |a|=√(a1^2+a2^2+a3^2).
- a·b = |a||b|cosθ; a×b is perpendicular to both, with |a×b|=|a||b|sinθ and direction by the right-hand rule.
- Collinear vectors satisfy b=λa; coplanar vectors satisfy scalar triple product [a b c]=0.
- Line through point a with direction b: r=a+λb. Plane through point a with normal n: (r-a)·n=0, or r·n=d.
- Angle between line direction b and plane normal n uses sinθ=|b·n|/(|b||n|); angle between two planes uses their normals.
Worked example
Find the plane through (1,2,3) with normal n=2i-j+4k. For a general point (x,y,z), (r-a)·n=0 gives 2(x-1)-1(y-2)+4(z-3)=0. Simplify: 2x-2-y+2+4z-12=0, hence 2x-y+4z-12=0.
Mark-losing traps
- Using the cross product when a scalar projection or angle is required.
- Confusing direction ratios of a line with the coordinates of a point on it.
- Taking the angle between a line and a plane as the angle between the line and the normal rather than its complement.
- Sign errors when expanding (r-a)·n=0.
- Forgetting that a normal vector is not unique; any nonzero multiple works.