Linear Independence Calculator

Step-by-Step Guide to Using the Linear Independence Calculator

Welcome to the Linear Independence Calculator. This guide will walk you through the steps of using this tool to determine whether two vectors are linearly independent, calculate the determinant, and find the angle between them. Follow each step carefully to ensure accurate results.

Step 1: Select the Vector Dimension

• Open the calculator interface: You will see a field labeled ‘Vector Dimension’.
• Choose the appropriate dimension: Use the dropdown menu to select either 2D (x, y) or 3D (x, y, z), depending on the dimensionality of the vectors you wish to analyze.

Step 2: Enter the Components of the Vectors

• For 2D vectors: You need to enter the x and y components for two vectors.
  • In the field labeled ‘Vector 1 – x component’, enter the x value of the first vector.
  • In the field labeled ‘Vector 1 – y component’, enter the y value of the first vector.
  • Similarly, fill in the fields for ‘Vector 2 – x component’ and ‘Vector 2 – y component’ with the components of the second vector.
• For 3D vectors: You must enter the x, y, and z components for two vectors.
  • Fill in the fields for ‘Vector 1 – x component’, ‘Vector 1 – y component’, and ‘Vector 1 – z component’ for the first vector.
  • Enter the components in ‘Vector 2 – x component’, ‘Vector 2 – y component’, and ‘Vector 2 – z component’ for the second vector.

Step 3: View the Results

• Determinant: After entering all required information, the calculator will compute the determinant of the vectors as per the dimension specified.
  • This is automatically calculated using the formula specific to the vector dimension you selected.
• Linear Independence: The calculator will analyze the determinant value to determine linear independence.
  • If the determinant is greater than a threshold (0.0001), it indicates that the vectors are linearly independent.
  • If not, the vectors are linearly dependent.
• Angle Between Vectors: The calculator also computes the angle between the two vectors.
  • The angle is given in degrees, rounded to two decimal places, providing a clear measure of the orientation between the vectors.

Follow these steps each time you use the Linear Independence Calculator to ensure accurate analysis of your vectors. Pay special attention to the units and ensure all necessary fields are completed before interpreting the results.