Questions: Find the cross products u × v and v × u for the vectors u = ⟨3,4,-9⟩ and v = ⟨1,1,-1⟩.

Solution Steps

To find the cross products \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) for the given vectors \(\mathbf{u} = \langle 3, 4, -9 \rangle\) and \(\mathbf{v} = \langle 1, 1, -1 \rangle\), we can use the formula for the cross product of two vectors in three-dimensional space. The cross product \(\mathbf{a} \times \mathbf{b}\) for vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\) is given by:

\[ \mathbf{a} \times \mathbf{b} = \langle a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \rangle \]

We will apply this formula to compute \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\).

Given vectors: \[ \mathbf{u} = \langle 3, 4, -9 \rangle \] \[ \mathbf{v} = \langle 1, 1, -1 \rangle \]

Using the cross product formula: \[ \mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle \] Substitute the values: \[ \mathbf{u} \times \mathbf{v} = \langle 4 \cdot (-1) - (-9) \cdot 1, (-9) \cdot 1 - 3 \cdot (-1), 3 \cdot 1 - 4 \cdot 1 \rangle \] Simplify: \[ \mathbf{u} \times \mathbf{v} = \langle -4 + 9, -9 + 3, 3 - 4 \rangle \] \[ \mathbf{u} \times \mathbf{v} = \langle 5, -6, -1 \rangle \]

Using the cross product formula: \[ \mathbf{v} \times \mathbf{u} = \langle v_2 u_3 - v_3 u_2, v_3 u_1 - v_1 u_3, v_1 u_2 - v_2 u_1 \rangle \] Substitute the values: \[ \mathbf{v} \times \mathbf{u} = \langle 1 \cdot (-9) - (-1) \cdot 4, (-1) \cdot 3 - 1 \cdot (-9), 1 \cdot 4 - 1 \cdot 3 \rangle \] Simplify: \[ \mathbf{v} \times \mathbf{u} = \langle -9 + 4, -3 + 9, 4 - 3 \rangle \] \[ \mathbf{v} \times \mathbf{u} = \langle -5, 6, 1 \rangle \]

Final Answer