Volume of Parallelepiped using Scalar Triple Product | Class 12

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What is the Volume of a Parallelepiped using Scalar Triple Product?

Grade Level:

Class 12

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Definition

What is it?

The volume of a parallelepiped can be found using the Scalar Triple Product. This product takes three vectors representing the co-terminal edges of the parallelepiped and gives a single scalar value, which is its volume. It essentially calculates the 'space' occupied by the 3D shape formed by these vectors.

Simple Example

Quick Example

Imagine you have three sticks of different lengths, meeting at one corner of a room, forming the edges of a box (a parallelepiped). If you know the exact direction and length of each stick (represented as vectors), the Scalar Triple Product helps you calculate how much air that box would hold. It's like finding the capacity of a water tank, but for a tilted box.

Worked Example

Step-by-Step

Let the three co-terminal edges of a parallelepiped be represented by vectors a = (1, 2, 3), b = (0, 1, 2), and c = (3, 0, 1).
---Step 1: Write the vectors in determinant form. The Scalar Triple Product [a b c] is calculated as the determinant of the matrix formed by these vectors.
| 1 2 3 |
| 0 1 2 |
| 3 0 1 |
---Step 2: Calculate the determinant. Expand along the first row.
Volume = 1 * ( (1 * 1) - (2 * 0) ) - 2 * ( (0 * 1) - (2 * 3) ) + 3 * ( (0 * 0) - (1 * 3) )
---Step 3: Simplify the terms inside the parentheses.
Volume = 1 * (1 - 0) - 2 * (0 - 6) + 3 * (0 - 3)
---Step 4: Perform the multiplications.
Volume = 1 * (1) - 2 * (-6) + 3 * (-3)
---Step 5: Continue simplifying.
Volume = 1 + 12 - 9
---Step 6: Calculate the final sum.
Volume = 13 - 9 = 4
---Step 7: Take the absolute value, as volume cannot be negative.
Volume = |4| = 4 cubic units.
Answer: The volume of the parallelepiped is 4 cubic units.

Why It Matters

Understanding parallelepiped volumes is crucial in fields like Engineering for designing structures, and in Physics for calculating forces and work done in 3D space. Even in AI/ML, these concepts help in understanding data distribution in multi-dimensional spaces. Engineers use this to ensure buildings are stable, and scientists use it to model complex systems.

Common Mistakes

MISTAKE: Forgetting to take the absolute value of the determinant. | CORRECTION: Volume is a scalar quantity and must always be positive. If your determinant calculation results in a negative number, always take its absolute value to get the correct volume.

MISTAKE: Mixing up the order of operations or signs when calculating the determinant. | CORRECTION: Carefully follow the rules for calculating a 3x3 determinant, paying close attention to the alternating positive and negative signs for each term in the expansion.

MISTAKE: Using vectors that are not co-terminal (not starting from the same point). | CORRECTION: Ensure the three given vectors represent edges that meet at a single common vertex of the parallelepiped. If they don't, you might need to adjust them by translation.

Practice Questions

Try It Yourself

QUESTION: Find the volume of the parallelepiped whose co-terminal edges are a = (2, -1, 3), b = (1, 0, 2), and c = (0, 3, 1). | ANSWER: 13 cubic units

QUESTION: If the volume of a parallelepiped with co-terminal edges p = (1, k, 2), q = (3, -1, 0), and r = (0, 1, -1) is 10 cubic units, find the possible value(s) of k. | ANSWER: k = 2 or k = -8/3

QUESTION: Three vectors are given as u = i + 2j - k, v = 3i - j + 2k, and w = i + 3j + 4k. Calculate the volume of the parallelepiped formed by these vectors. | ANSWER: 52 cubic units

MCQ

Quick Quiz

Which of the following is NOT a property of the Scalar Triple Product [a b c]?

[a b c] = [b c a]

[a b c] = -[b a c]

[a b c] = a . (b x c)

[a b c] = a x (b . c)

The Correct Answer Is:

Option D is incorrect because the dot product of a vector and a scalar (b . c is a scalar) is not defined as a vector operation. The Scalar Triple Product is correctly defined as a . (b x c).

Real World Connection

In the Real World

In India, architects and civil engineers use these vector concepts when designing complex building structures or bridges. For instance, when planning the layout of a multi-story building in a crowded city like Mumbai, they might use vectors to represent structural beams and columns. Calculating volumes helps ensure efficient use of space and material, crucial for projects like the new metro lines being built across various cities.

Key Vocabulary

Key Terms

Parallelepiped: A 3D shape similar to a cuboid, but its faces are parallelograms, not necessarily rectangles. | Scalar Triple Product: A mathematical operation involving three vectors that results in a scalar (single number) value. | Co-terminal Edges: Three edges of a 3D shape that meet at a single common vertex (corner). | Determinant: A special number calculated from a square matrix, used in solving systems of linear equations and finding volumes. | Vector: A quantity having both magnitude (size) and direction.

What's Next

What to Learn Next

Great job understanding parallelepiped volumes! Next, you can explore the 'Volume of a Tetrahedron using Scalar Triple Product'. This builds directly on what you've learned, showing how to find the volume of a triangular pyramid using a similar vector method. It's another exciting step in mastering 3D geometry!