Section 12.4: Cross Product

If we apply Theorems 8 and 9 to the standard basis vectors i, j, and k using \(\theta = \pi/2\), we obtain \[ \mathbf{i} \times \mathbf{j} = \mathbf{k} \qquad \mathbf{j} \times \mathbf{k} = \mathbf{i} \qquad \mathbf{k} \times \mathbf{i} = \mathbf{j} \] \[ \mathbf{j} \times \mathbf{i} = -\mathbf{k} \qquad \mathbf{k} \times \mathbf{j} = -\mathbf{i} \qquad \mathbf{i} \times \mathbf{k} = -\mathbf{j} \] Observe that \[ \mathbf{i} \times \mathbf{j} \ne \mathbf{j} \times \mathbf{i} \] Thus the cross product is not commutative. Also \[ \mathbf{i} \times (\mathbf{i} \times \mathbf{j}) = \mathbf{i} \times \mathbf{k} = -\mathbf{j} \] whereas \[ (\mathbf{i} \times \mathbf{i}) \times \mathbf{j} = \mathbf{0} \times \mathbf{j} = \mathbf{0} \] So the associative law for multiplication does not usually hold; that is, in general, \[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \ne \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \] However, some of the usual laws of algebra do hold for cross products. The following theorem summarizes the properties of vector products.

Properties of the Cross Product 11 If a, b, and c are vectors and \(c\) is a scalar, then 1. \(\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}\) 2. \((c\mathbf{a}) \times \mathbf{b} = c(\mathbf{a} \times \mathbf{b}) = \mathbf{a} \times (c\mathbf{b})\) 3. \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\) 4. \((\mathbf{a} + \mathbf{b}) \times \mathbf{c} = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}\) 5. \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}\) 6. \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\)

These properties can be proved by writing the vectors in terms of their components and using the definition of a cross product. We give the proof of Property 5 and leave the remaining proofs as exercises.

PROOF OF PROPERTY 5 If \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle, \mathbf{b} = \langle b_1, b_2, b_3 \rangle,\) and \(\mathbf{c} = \langle c_1, c_2, c_3 \rangle\), then \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1) \tag{12} \] \[ = a_1b_2c_3 - a_1b_3c_2 + a_2b_3c_1 - a_2b_1c_3 + a_3b_1c_2 - a_3b_2c_1 \] \[ = (a_2b_3 - a_3b_2)c_1 + (a_3b_1 - a_1b_3)c_2 + (a_1b_2 - a_2b_1)c_3 \] \[ = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \]