Vector Cross Product Calculator Form
How to Use This Vector Cross Product Calculator
This vector cross product calculator allows you to calculate the cross product of two vectors in 3D space. Follow the steps below to use the calculator effectively.
Step 1: Enter the Components of Vector A
In the form, you will see input fields for Vector A’s components. Enter the values for the X, Y, and Z components of Vector A. For example, if Vector A is \( \vec{A} = (3, 4, 2) \), input 3 in the \( A_x \) component field, 4 in the \( A_y \) component field, and 2 in the \( A_z \) component field.
Step 2: Enter the Components of Vector B
Next, enter the values for the X, Y, and Z components of Vector B in the corresponding input fields. For example, if Vector B is \( \vec{B} = (1, 5, 3) \), input 1 in the \( B_x \) component field, 5 in the \( B_y \) component field, and 3 in the \( B_z \) component field.
Step 3: Click the “Calculate Cross Product” Button
Once you have entered all the components for both vectors, click the “Calculate Cross Product” button. The calculator will automatically compute the cross product of the two vectors based on the values you provided.
Step 4: View the Result
The result will be displayed in the output field as a new vector. This is the cross product of the two vectors you entered. The result will be given in the format \( \vec{C} = (C_x, C_y, C_z) \), where \( C_x \), \( C_y \), and \( C_z \) are the respective components of the cross product vector.
Examples of Using the Vector Cross Product Calculator
Example 1: \( \vec{A} = (2, 3, 4) \) and \( \vec{B} = (1, 0, 5) \)
Step 1: Enter the values for \( \vec{A} = (2, 3, 4) \) and \( \vec{B} = (1, 0, 5) \).
Step 2: Click “Calculate Cross Product”.
Result: The cross product is \( \vec{C} = (15, -6, -3) \).
Example 2: \( \vec{A} = (1, 2, 3) \) and \( \vec{B} = (4, 5, 6) \)
Step 1: Enter the values for \( \vec{A} = (1, 2, 3) \) and \( \vec{B} = (4, 5, 6) \).
Step 2: Click “Calculate Cross Product”.
Result: The cross product is \( \vec{C} = (-3, 6, -3) \).
Example 1: Calculating the Cross Product of Vectors \( \vec{A} = (2, 3, 4) \) and \( \vec{B} = (1, 0, -1) \)
Let’s calculate the cross product of \( \vec{A} = (2, 3, 4) \) and \( \vec{B} = (1, 0, -1) \) step by step.
Step 1: Write Down the Vectors
We are given two vectors: \[ \vec{A} = (2, 3, 4), \quad \vec{B} = (1, 0, -1) \]
Step 2: Use the Cross Product Formula
The cross product of two vectors \( \vec{A} = (A_x, A_y, A_z) \) and \( \vec{B} = (B_x, B_y, B_z) \) is calculated using the determinant of a matrix: \[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} \] In our case, \( A_x = 2 \), \( A_y = 3 \), \( A_z = 4 \), \( B_x = 1 \), \( B_y = 0 \), and \( B_z = -1 \).
Step 3: Compute the Determinant
Now, expand the determinant: \[ \vec{A} \times \vec{B} = \hat{i} \left(3 \times -1 – 4 \times 0 \right) – \hat{j} \left(2 \times -1 – 4 \times 1 \right) + \hat{k} \left(2 \times 0 – 3 \times 1 \right) \] \[ = \hat{i} (-3) – \hat{j} (-2 – 4) + \hat{k} (0 – 3) \] \[ = -3 \hat{i} + 6 \hat{j} – 3 \hat{k} \]
Step 4: Final Cross Product
The final cross product is: \[ \vec{A} \times \vec{B} = (-3, 6, -3) \]
Example 2: Calculating the Cross Product of Vectors \( \vec{A} = (3, -2, 5) \) and \( \vec{B} = (1, 4, -2) \)
Step 1: Write Down the Vectors
We are given two vectors: \[ \vec{A} = (3, -2, 5), \quad \vec{B} = (1, 4, -2) \]
Step 2: Use the Cross Product Formula
Again, we use the cross product formula: \[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -2 & 5 \\ 1 & 4 & -2 \end{vmatrix} \]
Step 3: Compute the Determinant
Now, expand the determinant: \[ \vec{A} \times \vec{B} = \hat{i} \left(-2 \times -2 – 5 \times 4 \right) – \hat{j} \left(3 \times -2 – 5 \times 1 \right) + \hat{k} \left(3 \times 4 – -2 \times 1 \right) \] \[ = \hat{i} (4 – 20) – \hat{j} (-6 – 5) + \hat{k} (12 + 2) \] \[ = -16 \hat{i} + 11 \hat{j} + 14 \hat{k} \]
Step 4: Final Cross Product
The final cross product is: \[ \vec{A} \times \vec{B} = (-16, 11, 14) \]