Vector Algebra and Three-Dimensional Geometry
Vector Algebra and Three-Dimensional Geometry tests vector operations, dot and cross products, projections, direction cosines, and equations of lines and planes. The exam frequently combines vectors with shortest distance or angle problems.
Core rules
For a=a₁i+a₂j+a₃k, |a|=√(a₁²+a₂²+a₃²); unit vector is a/|a|. Dot product a·b=|a||b|cosθ is a scalar; it is zero for perpendicular vectors. Cross product a×b=|a||b|sinθ n is a vector perpendicular to both; |a×b| gives the area of the parallelogram, and half of it gives the triangle area. Direction cosines l,m,n satisfy l²+m²+n²=1. A line is r=a+λb. A plane is r·n=d. The angle between lines uses their direction vectors; the angle between planes uses their normals. Skew lines distance is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.
Worked example
Let a=i+j+k and b=i−j+k. Then a×b=|[i j k; 1 1 1; 1 −1 1]|=i(1·1−1·(−1))−j(1·1−1·1)+k(1·(−1)−1·1)=2i+0j−2k. So |a×b|=√(4+0+4)=2√2. The parallelogram area is 2√2; the triangle area is √2.
Traps
Dot product returns a scalar, cross product a vector; do not add them. a×b=−b×a, so order affects direction. Direction ratios are not unique; direction cosines are. In plane equations, the normal vector is not a point on the plane. Check whether the question asks for angle or acute angle.