Write the cross product of (1,2,0) and (1,4,2)

The cross product of two vectors **A** = (1, 2, 0) and **B** = (1, 4, 2) is given by the determinant of the following matrix:

\[
\mathbf{A} \times \mathbf{B} = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 0 \\
1 & 4 & 2 \\
\end{vmatrix}
\]

This can be expanded as:

\[
\mathbf{A} \times \mathbf{B} = \mathbf{i} \begin{vmatrix} 2 & 0 \\ 4 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 0 \\ 1 & 2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix}
\]

Now, let's compute each of these determinants:

1. For \(\mathbf{i}\):
\[
\begin{vmatrix} 2 & 0 \\ 4 & 2 \end{vmatrix} = (2 \cdot 2) - (0 \cdot 4) = 4
\]

2. For \(\mathbf{j}\):
\[
\begin{vmatrix} 1 & 0 \\ 1 & 2 \end{vmatrix} = (1 \cdot 2) - (0 \cdot 1) = 2
\]

3. For \(\mathbf{k}\):
\[
\begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix} = (1 \cdot 4) - (2 \cdot 1) = 2
\]

So, putting it all together:

\[
\mathbf{A} \times \mathbf{B} = 4\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}
\]

Therefore, the cross product is:

\[
\mathbf{A} \times \mathbf{B} = (4, -2, 2)
\]

Would you like further details or clarifications?

Here are 5 related questions for you to explore:
1. What is the physical significance of the cross product in physics?
2. How do you find the magnitude of the cross product?
3. Can the cross product be used in higher dimensions?
4. How does the cross product differ from the dot product?
5. What is the geometric interpretation of the cross product?

**Tip**: The cross product is perpendicular to the plane formed by the two vectors, following the right-hand rule.

Learn how to compute the cross product of vectors (1,2,0) and (1,4,2) using determinants. This step-by-step solution covers key vector algebra concepts and is suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation