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

Transcript text: Find the cross products $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{u}$ for the vectors $\mathbf{u}=\langle 3,4,-9\rangle$ and $\mathbf{v}=\langle 1,1,-1\rangle$.

Solution

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 $\mathbf{v} \times \mathbf{u}$.

Step 1: Define the Vectors

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

Step 2: Compute the Cross Product $\mathbf{u} \times \mathbf{v}$

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: \[ \mathbf{u} \times \mathbf{v} = \langle -4 + 9, -9 + 3, 3 - 4 \rangle \] \[ \mathbf{u} \times \mathbf{v} = \langle 5, -6, -1 \rangle \]

Step 3: Compute the Cross Product $\mathbf{v} \times \mathbf{u}$

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: \[ \mathbf{v} \times \mathbf{u} = \langle -9 + 4, -3 + 9, 4 - 3 \rangle \] \[ \mathbf{v} \times \mathbf{u} = \langle -5, 6, 1 \rangle \]