Vector Algebra
Vector Algebra tests dot and cross products, projections, angles, areas, scalar triple products and coplanarity. The exam favours short computations that test whether you know which product produces a scalar and which produces a vector.
Core rules
For a=(a1,a2,a3) and b=(b1,b2,b3), a·b=a1b1+a2b2+a3b3=|a||b|cosθ. The projection of a on b is (a·b)/|b|. The cross product a×b is a vector perpendicular to both, with |a×b|=|a||b|sinθ equal to the area of the parallelogram. In components, a×b=(a2b3−a3b2, a3b1−a1b3, a1b2−a2b1). The scalar triple product a·(b×c) is the determinant with rows a,b,c; its absolute value is the volume of the parallelepiped. Three vectors are coplanar exactly when the scalar triple product is zero. Also a×b=−b×a and a×a=0.
Worked example
Show that a=(1,2,3), b=(2,3,4), c=(3,4,5) are coplanar. Compute a·(b×c)= |1 2 3; 2 3 4; 3 4 5| =1(15−16)−2(10−12)+3(8−9)=−1+4−3=0. Hence the vectors are coplanar. Indeed c=2b−a, which displays the dependence directly.
Traps
A zero dot product means perpendicular, not parallel; a zero cross product means parallel. Do not treat a×b as a scalar. Reversing two vectors in a triple product changes its sign. Coplanarity is tested by the scalar triple product, not by a·b.