Dot (Inner) product of two vectors Calculator

$\color{red} \bold{Related \space Calculators:}$
Cross-Product of two vectors
Magnitude of a Vector
Unit Vector
Projection on Vectors
Angle between two vectors
Vector triple product
Scalar triple product

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

• 1. Introduction to the Dot-Product of two vectors
• 2. What is the Formulae used ?
• 3. How do I calculate the Dot-Product of two vectors?
• 4. Why choose our Dot-Product of two vectors 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 Dot-Product of two vectors

The dot product emerges as a powerful chord in the symphony of vectors, uniting magnitude and direction. Join us on this journey as we demystify the dot product, exploring its definition, practical applications, and the simple steps to harmonize vectors in mathematics and beyond.
$\bold{Definition}$
The dot product, also known as the scalar product, is an operation that combines two vectors to produce a scalar. It quantifies the degree of alignment or opposition between vectors.

2. What is the Formulae used?

The formula for calculating the $\bold{Cross \space product}$ of two vectors is given by:
$\space \space \space \space$ $\color{black}\bold{ \vec{a} \space . \space \vec{b} = |\vec{a}|.|\vec{b}|.Cos(\theta)}$
$\bold{|\vec{a}| \space and \space|\vec{b}|}$ represents the $\bold{magnitude \space or \space lengths}$ of the vectors a and b respectively.
${\bold{\theta}}$ represents the $\bold{acute \space angle}$ between $\bold{\vec{a} \space and \space\vec{b}}$.

3. How do I calculate the Dot Product of two vectors?

$\bold{Step\space1:}$ Calculate the magnitude or length of $\bold{\vec{a} \space and \space\vec{b}}$.
$\bold{Step\space2:}$ Calculate the angle between given vectors.
$\bold{Step\space3:}$ Apply the above-given formula to find the magnitude of the dot product.

4. Why choose our Dot product of two vectors 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 Dot product of two vectors.

6. How to use this calculator

This calculator will help you find the two vectors' dot product.
In the given input boxes, you have to put the value of the ${\bold{\vec{a} \space and \space \vec{b}}}$.
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

$\bold{Question:1}$
Given $\bold{\vec{a}}$ = 2i + 1j + 3k, $\bold{\vec{b}}$ = 4i + 2j + 1k find the value of $\bold{\vec{a} \space . \space \vec{b}}$ and its magnitude.
$\bold{Solution:1}$
(2i + 1j + 3k).(4i + 2j + 1k) = (2)(4) + (1)(2) + (3)(1) = 13

$\bold{Question:2}$
Given $\bold{\vec{a}}$ = 3i - 1j + 2k, $\bold{\vec{b}}$ = 2i + 4j - 1k find the value of $\bold{\vec{b} \space . \space \vec{a}}$ and its magnitude.
$\bold{Solution:1}$
(2i + 4j - 1k).(3i - 1j + 2k) = (2)(3) + (4)(-1) + (-1)(2) = 0

8. Frequently Asked Questions (FAQs):-

Can the dot-product be negative?

Yes, the dot-product can be negative if vectors have a significant component-wise difference.

What does a dot-product of zero signify?

A dot-product of zero indicates that the vectors are orthogonal (perpendicular) to each other.

Is the dot-product commutative?

No, the dot-product is not commutative; changing the order of vectors affects the result.

Can the dot-product be applied to complex vectors?

Yes, the dot product extends to complex vectors in complex vector spaces.

Is the dot-product applicable to vectors of different dimensions?

No, the vectors must have the same dimension for a valid dot product.

9. What are the real-life applications?

In physics, the dot-product finds applications in calculating work done by a force (F) along a displacement vector (d). The dot-product F⋅d yields the scalar work done.

10. Conclusion

As we conclude our exploration of the dot product, we recognize its role as the maestro orchestrating vector interactions. Embrace its simplicity and significance in quantifying alignment, and witness how this concept resonates across fields from physics to computer graphics. Though rooted in basic mathematical operations, the dot product stands as a fundamental tool, providing insights into vector relationships in the diverse landscapes of mathematics and real-world applications.

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]

$\fcolorbox{black}{lightpink}{\color{blue}{Click here to Ask any Doubt}}$
Are you Stuck on homework, assignments, projects, quizzes, labs, midterms, or exams?
To get connected to our tutors in real time. Sign up and get registered with us.