If r = (4, -2, 2) and F = (4, 8, 4), what is r x F?
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Question**: If r = (4, -2, 2) and F = (4, 8, 4), what is r x F?
This text appears to be a mathematical question focusing on vector cross product. The query asks for the cross product (denoted as "x") of two vectors \( r \) and \( F \).
### Explanation of Vectors and the Cross Product:
Vectors are fundamental to the study of physics, engineering, and computer science. In three-dimensional space, a vector is typically represented as \((x, y, z)\), where \( x \), \( y \), and \( z \) are its components along the respective axes.
The cross product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), results in a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). The resulting vector \( \mathbf{c} \) can be found using the determinant of a matrix as follows:
\[
\mathbf{a} \times \mathbf{b} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}
\]
Expanding this determinant, we get:
\[
\mathbf{a} \times \mathbf{b} = \left((a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}\right)
\]
### Given Vectors:
- \( r = (4, -2, 2) \)
- \( F = (4, 8, 4) \)
### Calculating the Cross Product:
Using the formula above:
\[
\mathbf{r} \times \mathbf{F} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & -2 & 2 \\
4 & 8 &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff12ec146-5781-453b-8388-2e7cc9921f6c%2F1f2b57f0-0ebb-4071-987b-e510b6116e4b%2F799o958_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question**: If r = (4, -2, 2) and F = (4, 8, 4), what is r x F?
This text appears to be a mathematical question focusing on vector cross product. The query asks for the cross product (denoted as "x") of two vectors \( r \) and \( F \).
### Explanation of Vectors and the Cross Product:
Vectors are fundamental to the study of physics, engineering, and computer science. In three-dimensional space, a vector is typically represented as \((x, y, z)\), where \( x \), \( y \), and \( z \) are its components along the respective axes.
The cross product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), results in a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). The resulting vector \( \mathbf{c} \) can be found using the determinant of a matrix as follows:
\[
\mathbf{a} \times \mathbf{b} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}
\]
Expanding this determinant, we get:
\[
\mathbf{a} \times \mathbf{b} = \left((a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}\right)
\]
### Given Vectors:
- \( r = (4, -2, 2) \)
- \( F = (4, 8, 4) \)
### Calculating the Cross Product:
Using the formula above:
\[
\mathbf{r} \times \mathbf{F} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & -2 & 2 \\
4 & 8 &
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