Math Problem Statement

B=T×N=∣ijk−40160−160∣=(0⋅0−16⋅(−16))i−(−4⋅0−16⋅0)j+(−4⋅(−16)−0⋅0)k\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}

Solution

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

\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.

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Math Problem Analysis

Mathematical Concepts

Vector Algebra
Cross Product

Formulas

Cross product formula using determinants

Theorems

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Suitable Grade Level

College Level

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