how did it become negative j when u cross multiply it 

Let's clarify the calculation of the cross product step by step to address the sign confusion.

Given vectors \( \mathbf{RAM} = (20, 18, -10) \) and \( \mathbf{RAN} = (-10, 8, 15) \), the cross product is computed using the determinant of a matrix:

\[
\mathbf{RAM} \times \mathbf{RAN} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
20 & 18 & -10 \\
-10 & 8 & 15
\end{vmatrix}
\]

This determinant expands as follows:

\[
\mathbf{RAM} \times \mathbf{RAN} = \mathbf{i} \left( 18 \cdot 15 - (-10) \cdot 8 \right) - \mathbf{j} \left( 20 \cdot 15 - (-10) \cdot (-10) \right) + \mathbf{k} \left( 20 \cdot 8 - 18 \cdot (-10) \right)
\]

Let's calculate each component step by step:

### \(\mathbf{i}\) Component:

\[
\mathbf{i} \left( 18 \cdot 15 - (-10) \cdot 8 \right) = \mathbf{i} \left( 270 + 80 \right) = \mathbf{i} \times 350
\]

### \(\mathbf{j}\) Component:

The \(\mathbf{j}\) component has a negative sign in front of it by the rule of determinant expansion:

\[
- \mathbf{j} \left( 20 \cdot 15 - (-10) \cdot (-10) \right) = - \mathbf{j} \left( 300 - 100 \right) = - \mathbf{j} \times 200
\]

### \(\mathbf{k}\) Component:

\[
\mathbf{k} \left( 20 \cdot 8 - 18 \cdot (-10) \right) = \mathbf{k} \left( 160 + 180 \right) = \mathbf{k} \times 340
\]

So the resulting cross product vector is:

\[
\mathbf{RAM} \times \mathbf{RAN} = (350, -200, 340)
\]

This explains why the \( \mathbf{j} \) component has a negative sign: it's due to the determinant expansion rule, where the middle term (the \( \mathbf{j} \) component) always carries a negative sign.

Would you like to continue with the calculations, or do you have any more questions about the steps?

Here are some related questions you might consider:
1. What is the rule for expanding the determinant of a 3x3 matrix?
2. How can the cross product be interpreted geometrically in three-dimensional space?
3. Why does the middle term in a cross product carry a negative sign?
4. How do different coordinate systems affect the calculation of a cross product?
5. What are some practical applications of the cross product in physics and engineering?

**Tip:** Always be careful with the signs when expanding a determinant, especially for the middle (j) component, which carries a negative sign by convention.

Explore why the cross product sign changes with a detailed step-by-step calculation. This problem involves vectors, determinants, and the properties of cross products in three-dimensional space. Suitable for advanced high school students and anyone interested in vector mathematics.

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