$$
\mathbf{B} = \mathbf{T} \times \mathbf{N} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-4 & 0 & 16 \\
0 & -16 & 0
\end{vmatrix} = (0 \cdot 0 - 16 \cdot (-16)) \mathbf{i} - (-4 \cdot 0 - 16 \cdot 0) \mathbf{j} + (-4 \cdot (-16) - 0 \cdot 0) \mathbf{k}
$$

To compute the cross product $\mathbf{B} = \mathbf{T} \times \mathbf{N}$, we use the determinant of the matrix formed by the standard unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$, and the components of vectors $\mathbf{T}$ and $\mathbf{N}$:

\[
\mathbf{B} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-4 & 0 & 16 \\
0 & -16 & 0
\end{vmatrix}
\]

The determinant of this matrix is calculated by expanding along the first row:

\[
\mathbf{B} = \mathbf{i} \begin{vmatrix} 0 & 16 \\ -16 & 0 \end{vmatrix} - \mathbf{j} \begin{vmatrix} -4 & 16 \\ 0 & 0 \end{vmatrix} + \mathbf{k} \begin{vmatrix} -4 & 0 \\ 0 & -16 \end{vmatrix}
\]

Now, compute each 2x2 determinant:

\[
\begin{vmatrix} 0 & 16 \\ -16 & 0 \end{vmatrix} = (0 \cdot 0) - (16 \cdot (-16)) = 256
\]

\[
\begin{vmatrix} -4 & 16 \\ 0 & 0 \end{vmatrix} = (-4 \cdot 0) - (16 \cdot 0) = 0
\]

\[
\begin{vmatrix} -4 & 0 \\ 0 & -16 \end{vmatrix} = (-4 \cdot -16) - (0 \cdot 0) = 64
\]

Thus, the cross product is:

\[
\mathbf{B} = 256 \mathbf{i} - 0 \mathbf{j} + 64 \mathbf{k}
\]

So, the resulting vector $\mathbf{B}$ is:

\[
\mathbf{B} = 256 \mathbf{i} + 64 \mathbf{k}
\]

This simplifies to:

\[
\mathbf{B} = (256, 0, 64)
\]

Would you like more details or have any questions about this solution? Here are some questions you might consider next:

1. How do you find the cross product of two vectors in general?
2. What are some properties of the cross product?
3. How do you interpret the result of a cross product geometrically?
4. How can the cross product be used in physics?
5. Can the cross product be used for vectors in higher dimensions?

**Tip:** Remember that the cross product is only defined in three-dimensional space and results in a vector that is perpendicular to both original vectors.

Learn how to compute the cross product of vectors using determinants. This example involves vector algebra and demonstrates the calculation step-by-step with a detailed solution. Suitable for college-level students and those studying vector algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation