Vectors (other)

Beyond basic vector operations (addition, scalar multiplication, dot product), vectors can be used to solve geometric problems, represent lines and planes, and express geometric transformations. Vectors can also be decomposed into components, and used to find areas of shapes like triangles and parallelograms.

Important properties

• A vector can be represented as an ordered pair $(x, y)$ in the plane.

• The vector from point $A(x_1, y_1)$ to $B(x_2, y_2)$ is $\vec{AB} = (x_2 - x_1, y_2 - y_1)$.

• The area of a triangle with vertices $A$, $B$, $C$ can be found using vectors: $\frac{1}{2} |\vec{AB} \times \vec{AC}|$ (in 2D, this is the absolute value of the determinant).

• Vectors can describe translations and other geometric transformations.

• Collinearity and parallelism can be checked using vector multiples.