Analytic Geometry and Vectors
Core concepts
This domain blends coordinate geometry with vectors. You should know the distance and midpoint formulas, slope conditions for parallel and perpendicular lines, equations of lines in several forms, and the equations of circles and parabolas. For vectors, understand components, magnitude, addition, scalar multiplication, dot product, and the geometric meaning of each. The dot product u·v = |u||v|cos θ detects perpendicularity (dot product zero) and gives the angle between vectors; the cross product in three dimensions yields a perpendicular vector.
Traps to watch
- Perpendicular slopes multiply to -1; a vertical line has undefined slope, not zero.
- The circle equation (x - h)² + (y - k)² = r² has centre (h, k) with signs reversed.
- Distance from a point to a line uses the perpendicular, not a horizontal or vertical drop.
- Dot product is a scalar, not a vector.
- When finding the angle, ensure the vectors are placed tail to tail.
Rule of thumb
Perpendicularity: dot product zero. Parallelism: one vector is a scalar multiple of the other. For a point-to-line distance, use |Ax₀ + By₀ + C| / sqrt(A² + B²).
Worked example
Find the distance from (3, 4) to the line 3x + 4y - 10 = 0. Using the formula: |3(3) + 4(4) - 10| / sqrt(9 + 16) = |9 + 16 - 10| / 5 = 15/5 = 3. For vectors u = (1, 2) and v = (2, -1), u·v = 2 - 2 = 0, so they are perpendicular.