Given Vector A=6i+4j+3k and B=2i-3j-3k. Find A×B

The cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \), denoted as \( \mathbf{A} \times \mathbf{B} \), can be found using the determinant of a 3x3 matrix. The formula is:

\[
\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}
\]

Given:
\[
\mathbf{A} = 6\hat{i} + 4\hat{j} + 3\hat{k}, \quad \mathbf{B} = 2\hat{i} - 3\hat{j} - 3\hat{k}
\]

Substitute these into the determinant:

\[
\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 4 & 3 \\ 2 & -3 & -3 \end{vmatrix}
\]

Now, expanding the determinant:

\[
\mathbf{A} \times \mathbf{B} = \hat{i} \begin{vmatrix} 4 & 3 \\ -3 & -3 \end{vmatrix} - \hat{j} \begin{vmatrix} 6 & 3 \\ 2 & -3 \end{vmatrix} + \hat{k} \begin{vmatrix} 6 & 4 \\ 2 & -3 \end{vmatrix}
\]

Evaluate each 2x2 determinant:

\[
\mathbf{A} \times \mathbf{B} = \hat{i}[(4)(-3) - (3)(-3)] - \hat{j}[(6)(-3) - (3)(2)] + \hat{k}[(6)(-3) - (4)(2)]
\]

Simplify:

\[
\mathbf{A} \times \mathbf{B} = \hat{i}[-12 + 9] - \hat{j}[-18 - 6] + \hat{k}[-18 - 8]
\]

\[
\mathbf{A} \times \mathbf{B} = \hat{i}[-3] - \hat{j}[-24] + \hat{k}[-26]
\]

Thus:

\[
\mathbf{A} \times \mathbf{B} = -3\hat{i} + 24\hat{j} - 26\hat{k}
\]

The cross product is:

\[
\mathbf{A} \times \mathbf{B} = -3\hat{i} + 24\hat{j} - 26\hat{k}
\]

Would you like further details or have any questions?

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Here are five related questions:

1. What is the magnitude of the vector \( \mathbf{A} \times \mathbf{B} \)?
2. How does the cross product differ from the dot product of two vectors?
3. Can the cross product of two parallel vectors ever be non-zero?
4. What is the geometric interpretation of the cross product?
5. How can we find the area of a parallelogram using the cross product?

**Tip:** The cross product results in a vector perpendicular to both original vectors.

Learn how to find the cross product of two vectors A = 6i + 4j + 3k and B = 2i - 3j - 3k using determinant expansion. This solution involves key concepts from vector algebra, including the calculation of determinants. Suitable for high school and university students.

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