Multiplying Complex Numbers in Cartesian Form
Multiply the complex numbers \( (3 + 4i) \) and \( (2 – 3i) \) in Cartesian form. Show all steps.
Distributive Property in Cross Product Proofs
Prove that the cross product operation is distributive over vector addition. In other words, show that for any vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), the identity \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c})\) holds. Provide a clear proof using either numeric or symbolic steps.
Exploring Cross Product Properties
Let \(\mathbf{a} = (1, 2, 3)\), \(\mathbf{b} = (0, -1, 2)\), and \(\mathbf{c} = (3, 0, -1)\). (a) Compute the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that \(\mathbf{a} \times \mathbf{b} = – (\mathbf{b} \times \mathbf{a})\). (b) Demonstrate the distributive property by first computing \(\mathbf{a} \times (\mathbf{b} + \mathbf{c})\) and then verifying that it equals […]