if vector a = 2i+3j+4k and vector b = 7i+8j-6k find a x b

To find the cross product \(\mathbf{a} \times \mathbf{b}\) of two vectors \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{b} = 7\mathbf{i} + 8\mathbf{j} - 6\mathbf{k}\), you can use the following determinant formula:

\[
\mathbf{a} \times \mathbf{b} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & 3 & 4 \\
7 & 8 & -6
\end{vmatrix}
\]

This determinant expands as:

\[
\mathbf{a} \times \mathbf{b} = \mathbf{i} \left( 3 \cdot (-6) - 4 \cdot 8 \right) - \mathbf{j} \left( 2 \cdot (-6) - 4 \cdot 7 \right) + \mathbf{k} \left( 2 \cdot 8 - 3 \cdot 7 \right)
\]

Now, calculate each component:

- The \(\mathbf{i}\) component:
  \[
  3 \cdot (-6) - 4 \cdot 8 = -18 - 32 = -50
  \]

- The \(\mathbf{j}\) component:
  \[
  2 \cdot (-6) - 4 \cdot 7 = -12 - 28 = -40
  \]

- The \(\mathbf{k}\) component:
  \[
  2 \cdot 8 - 3 \cdot 7 = 16 - 21 = -5
  \]

Thus, the cross product \(\mathbf{a} \times \mathbf{b}\) is:

\[
\mathbf{a} \times \mathbf{b} = -50\mathbf{i} + 40\mathbf{j} - 5\mathbf{k}
\]

Or in vector form:

\[
\mathbf{a} \times \mathbf{b} = -50\mathbf{i} + 40\mathbf{j} - 5\mathbf{k}
\]

Would you like more details on any part of this solution or have any questions?

Here are some related questions to explore:

1. How do you find the magnitude of the cross product \(\mathbf{a} \times \mathbf{b}\)?
2. What is the physical interpretation of the cross product?
3. How do you verify that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\)?
4. What is the cross product of \(\mathbf{a}\) with itself?
5. How does the cross product relate to the area of a parallelogram formed by two vectors?

**Tip:** Remember that the cross product results in a vector that is perpendicular to the plane formed by the original vectors.

Learn how to find the cross product of vectors a = 2i + 3j + 4k and b = 7i + 8j - 6k using the determinant formula. This detailed solution covers the calculation step-by-step, including formulas and vector components. Suitable for advanced high school students and anyone studying vector algebra.

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