Vectors and their components

The direction angle of a vector is found by applying an inverse tangent to its components.

The magnitude and direction of a sum of two vectors are found by solving the triangle they form with the law of cosines and sines.

The components of a vector at 30, 45 or 60 degrees use exact values of the special angles.

A point that divides a segment in a ratio is the start point plus that fraction of the vector between the endpoints.

A tangent is perpendicular to the radius, which is a dot product of zero between the tangent vector and the radius vector.

The distance between complex numbers is the length of the vector between their points.

A complex number is a point and also a vector, and adding complex numbers adds the vectors.

A conic is defined by distances from foci, which are lengths of vectors from the foci to the point.

The magnitude and direction of a vector are its polar coordinates r and theta.

The law of cosines is the source of the dot product formula for the angle between two vectors.

The area of a triangle with sides a and b and included angle C is half the magnitude of the cross product, one half ab sin C.

The side lengths of a polygon on a grid are the lengths of the vectors between its vertices.

Vectors prove geometric claims in coordinates: a quadrilateral is a parallelogram when opposite side vectors are equal.

A dilation multiplies each position vector by the scale factor, as a scalar multiple scales a vector.

Opposite sides of a parallelogram are equal vectors, so the diagonals are the sum and difference of two adjacent side vectors.

Two lines are perpendicular exactly when their direction vectors have dot product zero, which is the same as slopes whose product is -1.

A midsegment of a triangle is half the vector of the side it is parallel to.

Scaling a drawing multiplies each length by the scale factor, as multiplying a vector by a number scales its components.