The image shows a question asking for orthogonal vectors that span the same plane as the given vectors $x$ and $y$. Specifically, the vectors are:

$x = \begin{pmatrix} 1 \\ 2 \\ 7 \end{pmatrix}, \quad y = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$

To find an orthogonal vector that spans the same plane as $x$ and $y$, we need to calculate the cross product of $x$ and $y$. The cross product of two vectors results in a vector that is orthogonal to both.

The formula for the cross product of two 3-dimensional vectors $\mathbf{a} = (a_1, a_2, a_3)$ and $\mathbf{b} = (b_1, b_2, b_3)$ is:

$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Let me calculate that for these vectors.The cross product of the vectors $x = \begin{pmatrix} 1 \\ 2 \\ 7 \end{pmatrix}$ and $y = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$ is:

$\mathbf{x} \times \mathbf{y} = \begin{pmatrix} 11 \\ 19 \\ -7 \end{pmatrix}$

This vector is orthogonal to both $x$ and $y$ and spans the same plane as the original vectors.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you verify that two vectors are orthogonal?
2. What is the significance of the cross product in vector calculus?
3. Can two vectors have more than one orthogonal vector?
4. How is the magnitude of the cross product related to the area spanned by two vectors?
5. Can you calculate the angle between the vectors $x$ and $y$?

Tip: The cross product is particularly useful in physics for finding torques and normal vectors in three-dimensional space!