Coordinate geometry, trigonometry and vectors
Draw the relationships and label the given conditions before selecting a formula. A circle equation identifies both its center and squared radius; changing a sign inside a coordinate shift changes the center. The distance formula follows from the Pythagorean theorem. A line’s slope is rise divided by run, while a vertical line needs separate treatment because the usual slope is undefined.
For plane vectors, add corresponding components and use the dot product to connect coordinates with angle information. A zero dot product for nonzero vectors indicates perpendicularity. The magnitude of a sum is not generally the sum of the magnitudes. A projection measures the component in a specified direction, not the full original vector.
Trigonometric equations require attention to the interval and quadrant. Knowing a sine value can allow multiple angles, so do not silently discard a second permitted solution. Use radians in standard calculus formulas unless a conversion is explicitly included. Distinguish a triangle’s side length, area and angle; an answer with the wrong dimension cannot solve the original geometric question.