Scalar Triple Product Calculator

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$\bold{Table \space of \space Content}$

• 1. Introduction to the Scalar Triple Product
• 2. What is the Formulae used ?
• 3. How do I calculate the Scalar Triple Product?
• 4. Why choose our Scalar Triple Product Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Scalar Triple Product

Embark on a journey into scalar triple products, a fundamental concept in vector algebra. This blog elucidates the definition, formula, and practical applications, providing a simplified guide for understanding and calculating scalar triple products.
$\bold{Definition}$
The scalar triple product involves three vectors, A, B, and C, and is calculated as the dot product of the cross product of A and B with vector C. It is denoted as $\bold{\vec{A}.(\vec{B} X \vec{C})}$

2. What is the Formulae used?

For $\vec{A}$ = $<a_1, a_2, a_3>$, $\vec{B}$ = $<b_1, b_2, b_3>$ and $\vec{C}$ = $<c_1, c_2, c_3>$, the scalar triple product is given by:
$\vec{A}.(\vec{B} X \vec{C})$ = $a_1.(b_2.c_3 - b_3.c_2) + a_2.(b_3.c_1 - b_1.c_3) + a_3.(b_1.c_2 - b_2.c_1)$
$\vec{A}.(\vec{B} X \vec{C})$ is also represented by [ $\vec{A} \space \vec{B} \space \vec{C}$ ] called as $\bold{Box \space product}$.
$\vec{A}.(\vec{B} X \vec{C})$ = $\vec{C}.(\vec{A} X \vec{B})$= $\vec{B}.(\vec{C} X \vec{A})$
Volume of Parallelopiped is also calculated by the scalar triple product of given three edge vectors. The vectors must be in three-dimensional space, and the scalar triple product is only defined for three vectors.

3. How do I calculate the Scalar triple product?

Determine the three vectors A, B, and C involved in the scalar triple product.
Calculate the cross product B × C.
Take the dot product of vector A with the result from the cross product.
Simplify the expression to obtain the scalar triple product.

4. Why choose our Scalar triple product calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on the concept of how to find the Scalar Triple product.

6. How to use this calculator

This calculator will help you to find the Scalar Triple Product.
In the given input boxes, you have to put the value of the coordinates of all the vectors.
After clicking on the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question:1}$
A=⟨2,−1,3⟩, B=⟨4,5,1⟩, C=⟨−3,2,6⟩:Calculate A⋅(B × C).
$\bold{Solution:1}$
B × C = ⟨29,−26,22⟩
A⋅(B×C) = 2(29)+(−1)(−26)+3(22) = 139
A⋅(B×C) = 139.

$\bold{Question:2}$
A=⟨1,2,-1⟩, B=⟨3,0,2⟩, C=⟨−2,1,4⟩:Calculate A⋅(B × C).
$\bold{Solution:2}$
B × C = ⟨8, −14, 3⟩
A⋅(B×C) = 1(8)+(2)(−14)+(-1)(3) = -27
A⋅(B×C) = -27.

8. Frequently Asked Questions (FAQs):-

What does the scalar triple product represent geometrically?

It represents the volume of the parallelepiped formed by the three vectors.

Can the order of vectors be changed in the scalar triple product?

Yes, changing the order changes the sign of the result.

Is the scalar triple product commutative?

No, A⋅(B×C) $\ne$ C⋅(B×A).

What happens when the scalar triple product is zero?

It indicates that the three vectors are coplanar.

Can scalar triple products be negative?

Yes, the result can be positive, negative, or zero, depending on the orientation of the vectors.

9. What are the real-life applications?

In physics, the scalar triple product finds applications in calculating work done by a force moving an object in a given direction, involving force vectors and displacement vectors.

10. Conclusion

Unraveling the scalar triple product opens doors to understanding spatial relationships and finding applications in various fields. Though rooted in vector algebra, this versatile concept proves invaluable in geometric interpretations and real-world problem-solving.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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