A vector in three-dimensional space is a triple \(\bm{v} = \bigl\langle v_1, v_2, v_3\bigr\rangle\) that, without any other context, denotes movement from the origin to the point \(\bigl(v_1, v_2, v_3\bigr).\) The numbers \(v_1\) and \(v_2\) and \(v_3\) are called the \(x\)-, \(y\)-, and \(z\)-components of the vector. Sometimes we talk about the vector from a point \(A\) to a point \(B,\) which we’ll denote \(\overrightarrow{AB}.\) The points \(A\) and \(B\) are referred to as the initial and terminal point respectively. Given two vectors (forces) \(\bm{u}\) and \(\bm{v}\) applied one after the other to a point (object), the resultant vector (force) is the sum \(\bm{u}+\bm{v}.\) If a vector (force) \(\bm{u}\) is scaled by a factor of \(k\) then \(k\bm{u}\) will denote the scaled vector. In terms of their components, \[ \bm{u} = \langle u_1, u_2, u_3 \rangle \quad \bm{v} = \langle v_1, v_2, v_3 \rangle \quad \implies \quad \bm{u} + \bm{v} = \langle u_1+v_1, u_2+v_2, u_3+v_3 \rangle \quad k\bm{u} = \langle k u_1, k u_2, k u_3 \rangle. \] The vector \(-\bm{v}\) will have the opposite direction as \(\bm{v}.\) The length of a vector \(\bm{v}\) is referred to as its magnitude (or sometimes its modulus), and is denoted \(|\bm{v}|.\) The magnitude of a vector \(\bm{v} = \langle v_1, v_2, v_3 \rangle\) can be calculated explicitly in terms of its components as \(\sqrt{v_1^2 + v_2^2 + v_3^2}.\) There is a unique vector with no magnitude, indicating no movement, called the zero vector, denoted \(\bm{0}.\) A unit vector is a vector of length one; given a vector \(\bm{v}\) the unit vector in the same direction as \(\bm{v}\) will be denoted \(\bm{\hat{v}}\) (“vee hat”) and calculated as \(\bm{\hat{v}} = \frac{1}{|\bm{v}|}\bm{v}.\) The unit coordinate vectors \({\mathbf{i} = \langle 1,0,0 \rangle}\) and \({\mathbf{j} = \langle 0,1,0 \rangle}\) and \({\mathbf{k} = \langle 0,0,1 \rangle}\) are the unit vectors pointing in the direction of each coordinate axis. As a matter of notation, we sometimes write \(\langle v_1,v_2,v_3 \rangle\) as \(v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}.\)
The “slope” of a plane and the “direction” of a line can more easily be described in terms of vectors. For a plane with equation \( {Ax + By + Cz = D\,,} \) the vector \(\bigl\langle A,B,C \bigr\rangle\) will be normal (perpendicular, sticking-straight-outta) that plane. A parametrically-defined line \[ x(t) = x_0 + At \qquad y(t) = y_0 + Bt \qquad z(t) = z_0 + Ct\] can also be described with a single “vector equation” \( {\bm{r}(t) = \bigl(x_0, y_0, z_0) + \bigl\langle A, B, C \bigr\rangle t.}\)