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The image shows a problem asking to calculate the cross product (vektorprodukt in Swedish) between two 3D vectors:

\[
\mathbf{a} = \begin{pmatrix} 3 \\ -10 \\ 11 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -5 \\ -12 \\ 11 \end{pmatrix}
\]

To compute the cross product \(\mathbf{a} \times \mathbf{b}\), we use the formula for the cross product of two vectors in \(\mathbb{R}^3\):

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

Where \(\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}\). The determinant expands to:

\[
\mathbf{a} \times \mathbf{b} = \left( \hat{i}(a_2 b_3 - a_3 b_2) - \hat{j}(a_1 b_3 - a_3 b_1) + \hat{k}(a_1 b_2 - a_2 b_1) \right)
\]

Let's substitute the values from \(\mathbf{a}\) and \(\mathbf{b}\) and compute the cross product.The cross product of the vectors \(\mathbf{a} = \begin{pmatrix} 3 \\ -10 \\ 11 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -5 \\ -12 \\ 11 \end{pmatrix}\) is:

\[
\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 22 \\ -88 \\ -86 \end{pmatrix}
\]

This is the vector result of the cross product. 

Do you want further details or have any questions?

Here are 5 related questions you could explore:
1. How do you calculate the dot product of two vectors?
2. What is the geometric interpretation of a cross product?
3. How does the cross product differ in 2D and 3D spaces?
4. How can you determine the area of a parallelogram using the cross product?
5. What is the significance of the direction of the cross product vector?

**Tip:** The magnitude of the cross product vector gives the area of the parallelogram formed by the two vectors.

Beräkna vektorprodukten mellan vektorerna (3, -10, 11) och (-5, -12, 11).

Learn how to calculate the cross product between the 3D vectors (3, -10, 11) and (-5, -12, 11) with a detailed solution. This problem covers essential vector algebra concepts, including the cross product formula in 3D space. Suitable for Grades 10-12.

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