Section 12.2: Vectors

Properties of Vectors If a, b, and c are vectors in \(V_n\) and \(c\) and \(d\) are scalars, then 1. \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\) 2. \(\mathbf{a} + (\mathbf{b} + \mathbf{c}) = (\mathbf{a} + \mathbf{b}) + \mathbf{c}\) 3. \(\mathbf{a} + \mathbf{0} = \mathbf{a}\) 4. \(\mathbf{a} + (-\mathbf{a}) = \mathbf{0}\) 5. \(c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}\) 6. \((c + d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}\) 7. \((cd)\mathbf{a} = c(d\mathbf{a})\) 8. \(1\mathbf{a} = \mathbf{a}\) —

These eight properties of vectors can be readily verified either geometrically or algebraically. For instance, Property 1 can be seen from Figure 4 (it’s equivalent to the Parallelogram Law) or as follows for the case \(n = 2\): \[ \mathbf{a} + \mathbf{b} = \langle a_1, a_2 \rangle + \langle b_1, b_2 \rangle = \langle a_1 + b_1, a_2 + b_2 \rangle \] \[ = \langle b_1 + a_1, b_2 + a_2 \rangle = \langle b_1, b_2 \rangle + \langle a_1, a_2 \rangle = \mathbf{b} + \mathbf{a} \] We can see why Property 2 (the associative law) is true by looking at Figure 16 and applying the Triangle Law several times: the vector \(\vec{PQ}\) is obtained either by first constructing \(\mathbf{a} + \mathbf{b}\) and then adding c or by adding a to the vector \(\mathbf{b} + \mathbf{c}\).