Vector Cross Product
Solve problem in this form \[ [13,4,15],[9,3,20], \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\]
Vector Cross Product
A vector cross product calculator computes the cross product of two three-dimensional vectors, resulting in a new vector that is perpendicular to both input vectors. It simplifies the calculation using the determinant method, providing the components of the resultant orthogonal vector quickly and accurately.
Lesson Note: Vector Cross Product
Introduction to Vector Cross Product
The cross product (or vector product) of two vectors is a binary operation in three-dimensional space. It results in a new vector that is perpendicular to the plane containing the original vectors. This operation is particularly useful in physics and engineering for finding a vector orthogonal to two given vectors.
Notation
If \(\vec{a}\) and \(\vec{b}\) are two vectors, their cross product is denoted by \(\vec{a} \times \vec{b}\).
Properties of Cross Product
1. Non-Commutative:
\[
\vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}
\]
In fact, \(\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})\).
2. Orthogonal Result:
The resulting vector is perpendicular to both \(\vec{a}\) and \(\vec{b}\).
3. Magnitude:
The magnitude of \(\vec{a} \times \vec{b}\) is given by:
\[
|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta
\]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).
Formula for Cross Product
If \(\vec{a} = \begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix}\), the cross product \(\vec{c} = \vec{a} \times \vec{b}\) is calculated as:
\[
\vec{c} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{vmatrix}
\]
This determinant can be expanded as:
\[
\vec{c} = \mathbf{i}(a_y b_z – a_z b_y) – \mathbf{j}(a_x b_z – a_z b_x) + \mathbf{k}(a_x b_y – a_y b_x)
\]
So,
\[
\vec{c} = \begin{pmatrix}
a_y b_z – a_z b_y \\
a_z b_x – a_x b_z \\
a_x b_y – a_y b_x
\end{pmatrix}
\]
Example
Let’s find the cross product of \(\vec{a} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}\).
Using the formula:
\[
\vec{a} \times \vec{b} = \begin{pmatrix}
3 \cdot 7 – 4 \cdot 6 \\
4 \cdot 5 – 2 \cdot 7 \\
2 \cdot 6 – 3 \cdot 5
\end{pmatrix} = \begin{pmatrix}
21 – 24 \\
20 – 14 \\
12 – 15
\end{pmatrix} = \begin{pmatrix}
-3 \\
6 \\
-3
\end{pmatrix}
\]
So, the cross product \(\vec{a} \times \vec{b} = \begin{pmatrix} -3 \\ 6 \\ -3 \end{pmatrix}\).
The vector cross product is an essential tool in 3D vector operations. It results in a vector orthogonal to the plane of the given vectors, with a magnitude dependent on the sine of the angle between them and the magnitudes of the original vectors. Mastery of this concept is crucial for applications in physics, engineering, and computer graphics.
Summary
– Cross product of two vectors results in a vector perpendicular to both.
– Formula: \(\vec{a} \times \vec{b} = \begin{pmatrix} a_y b_z – a_z b_y \\ a_z b_x – a_x b_z \\ a_x b_y – a_y b_x \end{pmatrix}\).
– Properties: Non-commutative, orthogonal result, magnitude given by \(|\vec{a}||\vec{b}|\sin\theta\).