Solving geometric problems

Understand that geometric problems often involve calculations with vectors. A vector is a quantity that has both magnitude and direction. Key properties to remember include vector addition, subtraction, scalar multiplication, and the geometric interpretations of these operations.
One common application of vectors in geometric problems is to calculate distances and midpoints between points. Understand that the distance between two points P(a,b) and Q(c,d) in a 2-dimensional plane is given by √[(c-a)²+(d-b)²], while the midpoint is given by ((a+c)/2, (b+d)/2).
Learn the method to calculate vector and scalar products. The scalar or dot product is a way of multiplying two vectors which results in a scalar. The vector or cross product is a way of multiplying two vectors which results in a vector.

Calculate the scalar product of two vectors, understand its relation to the angle between the vectors and use this information to find vector equations of planes. The angle θ between two vectors a and b can be found using the formula cos(θ) = (a . b) / (

), where “.” represents the dot product and

represents the magnitude of vector a.

Recognize the importance of proofs in geometry, and understand the main proof techniques including direct proof, indirect proof, and proof by exhaustion.
Become comfortable with using geometry software or graphing calculators to solve problems that involve manipulating or visualising geometric shapes.
Solve geometric problems that involve the use of trigonometry, especially problems related to circles, triangles, and polar coordinates. Important topics include the sine, cosine, and tangent ratios, the Pythagorean identity, the double-angle and half-angle formulas, and the conversion between rectangular and polar coordinates.
Work out problems involving complex numbers in the Argand diagram. Understand how to plot complex numbers, and how to multiply and divide them using trigonometric form.
Use differential and integral calculus in the context of geometric problems. For example, differentiation can be used to find tangents to curves, and integration can be used to find areas under curves and volumes of revolution.
Lastly, always check the answers for plausibility. Not all math problems have unique solutions, so it is important to verify the solutions in the context of the original problem.