Calculate volume using three vectors
Step 1: Understand Parallelepiped Structure
A parallelepiped is a 3D shape formed by three vectors from a common origin. It\'s like a skewed box where edges don\'t necessarily meet at right angles.
Why: Visualizing the shape helps you correctly identify and enter the three edge vectors.
Step 2: Identify Your Three Vectors
Determine vectors A, B, and C representing the three edges from the parallelepiped\'s base point. Each vector has x, y, z components.
Why: The scalar triple product formula A¡(B\u00d7C) requires all three vectors; missing or incorrect vectors invalidate the result.
Step 3: Enter Vector A (x, y, z)
Input the three Cartesian components of the first edge vector A into the corresponding x, y, and z fields.
Why: Each component contributes to the cross product and must be accurate; a sign error changes the volume sign (absolute value handles this).
Step 4: Enter Vectors B and C
Input the x, y, z components of vectors B and C in the same manner. Verify order: vectors should form a right-handed system conceptually.
Why: The scalar triple product is order-dependent; permuting vectors changes the sign (which is why we take absolute value).
Step 5: Calculate and Validate Result
Click Calculate Volume. The result is |A¡(BĂC)|. Verify by checking units (should be cubic units) and comparing to expected size.
Why: Validation catches if you entered unit vectors vs. scaled vectors, or if components are in different measurement systems.
Scenario:
Computing the volume of a parallelepiped with vectors A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1).
Step 1 â Recognize Shape:
These three orthogonal vectors form a unit cube (length 1 in each direction, all perpendicular).
Step 2 â Identify Components:
A has components (1, 0, 0), B has (0, 1, 0), C has (0, 0, 1). All components are explicit and ready to enter.
Step 3 â Input Vector A:
Enter x=1, y=0, z=0 into the Vector A fields.
Step 4 â Input Vectors B and C:
Enter B as (0, 1, 0) and C as (0, 0, 1). All three vectors are now defined.
Step 5 â Apply Scalar Triple Product:
V = |A¡(B\u00d7C)| = |(1,0,0)¡((0,1,0)\u00d7(0,0,1))| = |(1,0,0)¡(1,0,0)| = |1| = 1.
Verification:
For a unit cube, volume should be 1 cubic unit. Result matches expected value. â
Result & Interpretation:
Volume = 1 cubic unit. This is the simplest parallelepiped; scaling any vector by 2 would scale volume by 2, making a 2\u00d71\u00d71 box.
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