Vectors in geometry

A vector has both length and direction. In component form, a vector is written as \(\langle x, y \rangle\) or \(x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) points right and \(\mathbf{j}\) points up. To add vectors, add their corresponding components: \(\langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle\). To multiply a vector by a scalar \(k\), multiply each component: \(k\langle x, y \rangle = \langle kx, ky \rangle\). To find a vector from an initial point \(P(x_1, y_1)\) to a terminal point \(Q(x_2, y_2)\), subtract the initial coordinates from the terminal coordinates: \(\vec{PQ} = \langle x_2-x_1, y_2-y_1 \rangle\).