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What is the formula for calculating the cross product of two vectors?
What is the formula for calculating the cross product of two vectors?
The cross product of two vectors $oldsymbol{A}$ and $oldsymbol{B}$ is given by $oldsymbol{A} imes oldsymbol{B} = |oldsymbol{A}| |oldsymbol{B}| ext{sin}( heta) oldsymbol{n}$, where $ heta$ is the angle between the vectors and $oldsymbol{n}$ is the unit vector perpendicular to the plane containing the vectors.
Calculate the cross product $oldsymbol{A} imes oldsymbol{B}$ using the components of both vectors.
Calculate the cross product $oldsymbol{A} imes oldsymbol{B}$ using the components of both vectors.
$oldsymbol{A} imes oldsymbol{B} = (3 - 1)oldsymbol{i} + (1 - 1)oldsymbol{j} + (1 - 3)oldsymbol{k} = 2oldsymbol{i} - 2oldsymbol{k}$
In the problem, what are the components of vector $oldsymbol{A}$?
In the problem, what are the components of vector $oldsymbol{A}$?
The components of vector $oldsymbol{A}$ are 1 in the $oldsymbol{i}$ direction, 3 in the $oldsymbol{j}$ direction, and 1 in the $oldsymbol{k}$ direction.
What are the components of vector $oldsymbol{B}$?
What are the components of vector $oldsymbol{B}$?
What is the result of the cross product $oldsymbol{A} imes oldsymbol{B}$?
What is the result of the cross product $oldsymbol{A} imes oldsymbol{B}$?
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Study Notes
Cross Product
- The question asks for the cross product of two vectors, $\vec{A}$ and $\vec{B}$
- The cross product is a vector operation that results in a vector perpendicular to both input vectors
- It can be calculated using the determinant of a matrix
Vector Representation
- $\vec{A}$ and $\vec{B}$ are represented in terms of the standard unit vectors $\vec{i}$, $\vec{j}$, and $\vec{k}$
- $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$, and $\vec{k} = (0, 0, 1)$
The Cross Product Calculation
- The cross product of $\vec{A}$ and $\vec{B}$ can be calculated using the determinant of a matrix:
- $\vec{A} \times \vec{B} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 3 & 1 \\ 1 & 1 & 1 \end{vmatrix}$
- Expanding the determinant results in:
- $\vec{A} \times \vec{B} = (3-1) \vec{i} - (1-1) \vec{j} + (1-3) \vec{k}$
- $\vec{A} \times \vec{B} = 2 \vec{i} - 2 \vec{k}$
Result
- The cross product of $\vec{A}$ and $\vec{B}$ is $2 \vec{i} - 2 \vec{k}$
- This represents a vector perpendicular to both $\vec{A}$ and $\vec{B}$
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