Coordinate geometry and vectors
Coordinate geometry and vectors in SEE Optional Mathematics tests algebraic representation of points, lines, and directed segments, and vector operations in two dimensions.
Coordinate geometry
Distance between A(x₁,y₁) and B(x₂,y₂) is √((x₂−x₁)²+(y₂−y₁)²). Midpoint is ((x₁+x₂)/2,(y₁+y₂)/2); section formula for ratio m:n is ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)). Slope m=(y₂−y₁)/(x₂−x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂=−1. Line equation: y−y₁=m(x−x₁); general form ax+by+c=0. Area of triangle with vertices is ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.
Vectors
A vector has magnitude and direction. In components, a=(a₁,a₂), b=(b₁,b₂): a+b=(a₁+b₁,a₂+b₂), ka=(ka₁,ka₂). Magnitude |a|=√(a₁²+a₂²). Unit vector â=a/|a|. Dot product a·b=a₁b₁+a₂b₂=|a||b|cosθ; perpendicular vectors have dot product zero. Position vector of P is OP.
Worked example
Find the equation of the line through (1,2) and (3,6). Slope=(6−2)/(3−1)=2. Using y−2=2(x−1), y=2x. Check at x=3: y=6.
Traps
Reversing section-formula weights; forgetting absolute value in area; treating vector magnitude as a signed number; using the dot-product cosine formula with degrees and radians mixed.