Coordinate Geometry and Vectors
Coordinate geometry and vectors tests distance, midpoint, slope, straight-line equations, and vector operations in two dimensions. SEE items may ask for a length, a midpoint, a line equation, or the magnitude and direction of a vector.
Core rules
- Distance between (x₁,y₁) and (x₂,y₂): √[(x₂−x₁)²+(y₂−y₁)²].
- Midpoint: ((x₁+x₂)/2,(y₁+y₂)/2). Section formula for internal division in ratio m:n: ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).
- Slope m=(y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
- Line equation: y−y₁=m(x−x₁); slope-intercept y=mx+c.
- Vector a=(a₁,a₂) has magnitude |a|=√(a₁²+a₂²). Addition and scalar multiplication are componentwise. Position vector of P is OP.
Worked example
Find the distance and midpoint of A(1,2) and B(4,6). Distance=√[(4−1)²+(6−2)²]=√(9+16)=√25=5. Midpoint=((1+4)/2,(2+6)/2)=(2.5,4). The slope of AB=(6−2)/(4−1)=4/3, so a line perpendicular to AB has slope −3/4, not 3/4. For a=(3,4), |a|=√(9+16)=5.
Traps
- Subtracting coordinates in the wrong order, which flips the slope sign.
- Using the reciprocal instead of the negative reciprocal for a perpendicular slope.
- Mixing up internal and external section formulas.
- Forgetting to take the square root in vector magnitude.
- Writing a vector as a point, or omitting direction when it is requested.