To define two vectors $\mathbf{u}$ and $\mathbf{v}$ with a cross product $\mathbf{w} = (27, -8, 14)$, we need to find vectors such that their cross product gives the vector $\mathbf{w}$.

The cross product of two vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ in three-dimensional space is given by the determinant of the following matrix:

$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$

This results in the vector:

$\mathbf{w} = \left( u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \right)$

We want this to equal $\mathbf{w} = (27, -8, 14)$. To construct such vectors, we can either solve this system directly or choose arbitrary vectors $\mathbf{u}$ and $\mathbf{v}$ that satisfy this condition.

Example:

Let’s assume $\mathbf{u} = (3, 5, 2)$ and find $\mathbf{v}$.

The cross product is calculated as:

= \left( 5v_3 - 2v_2, 2v_1 - 3v_3, 3v_2 - 5v_1 \right)$$ Equating this to $$(27, -8, 14)$$, we get the system: 1. $$5v_3 - 2v_2 = 27$$ 2. $$2v_1 - 3v_3 = -8$$ 3. $$3v_2 - 5v_1 = 14$$ Now, solving this system will give $$\mathbf{v}$$. Would you like me to proceed with solving this, or would you prefer to choose different vectors for $$\mathbf{u}$$ and $$\mathbf{v}$$? ### Questions for further exploration: 1. What are the conditions for two vectors to have a non-zero cross product? 2. How does the magnitude of the cross product relate to the area of the parallelogram formed by two vectors? 3. Can the cross product be zero for non-zero vectors? 4. What is the geometric interpretation of the direction of the cross product vector? 5. How does the right-hand rule apply to the cross product of two vectors? ### Tip: The cross product of two vectors is always perpendicular to the plane containing the two vectors.