Vectors
Vectors covers vector operations, position vectors, inner products, equations of lines and planes, and geometric applications. Course 2 questions often require translating perpendicularity or parallelism into dot-product or scalar-multiple equations.
Core rules
- Vector addition is componentwise; scalar multiplication changes length, and a negative scalar reverses direction.
- Position vector of point A is OA. For points A and B, AB=b-a.
- Inner product: a·b=|a||b|cosθ=a_1b_1+a_2b_2. Perpendicular vectors have a·b=0.
- Line through A with direction d: x=a+td. Plane through A with normal n: n·(x-a)=0.
- The projection of a onto b is ((a·b)/(b·b))b, useful for foot-of-perpendicular problems.
Worked example
Let a=(1,2) and b=(3,-1). Then a·b=1·3+2·(-1)=1. Since |a|=√5 and |b|=√10, cosθ=1/√50=1/(5√2). If a line has direction (3,4) and passes through (1,2), its parametric equation is (x,y)=(1,2)+t(3,4).
Traps
Do not treat a dot product as a vector-it is a scalar. Do not divide by a vector. In a line equation, the direction vector cannot be zero. For planes, the normal vector is perpendicular to every direction vector in the plane. Check whether a problem uses 2D or 3D coordinates before writing equations.