Area of a triangle formed by two coincident Vectors

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

• 1. Introduction to the Area of the triangle formed by coincident vectors
• 2. What is the Formulae used ?
• 3. How do I calculate the Area of the triangle formed by coincident vectors?
• 4. Why choose our Area of the triangle formed by coincident 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 Area of the triangle formed by coincident vectors: -

Embarking on a geometric journey, this blog unravels the fascinating concept of finding the area of a triangle formed by two coincident vectors. Beyond the equations, we explore the practicality and applications of this mathematical gem.
$\bold{Definition}$
In the realm of vectors, when two vectors are coincident (parallel and have the same direction), they give rise to a unique triangle. The area of this triangle becomes a captivating aspect of vector mathematics.

2. What is the Formulae used?

The formula for calculating the $\bold{Area \space of \space triangle}$ by two coincident vectors $\vec{a}$ & $\vec{b}$ is given by:
$\color{black}\bold{Area \space = (\frac{1}{2}).|\vec{a} \space x \space \vec{b}| \space or \space (\frac{1}{2}).|\vec{b} \space x \space \vec{a}|} = (\frac{1}{2}).|\vec{a}|.|\vec{b}|.sin(\theta)$ $\bold{(\frac{1}{2}).|\vec{a} \space x \space \vec{b}| = (\frac{1}{2}).|\begin{vmatrix} \^{i} & \^{j} & \^{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}}|$
where,
$\bold{|\vec{a}| \space and \space|\vec{b}|}$ represents the $\bold{magnitude \space or \space lengths}$ of the vectors a and b respectively. ${\bold{\theta}}$ represents the $\bold{acute \space angle}$ between $\bold{\vec{a} \space and \space\vec{b}}$. This formula is applicable only when the vectors A and B are coincident.

3. How do I calculate the Area of the triangle formed by coincident 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 cross product.
$\bold{Step\space4:}$ We can apply the determinant method directly to find the cross product of two vectors.

4. Why choose our Area of the triangle formed by the coincident 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 Area of the triangle formed by coincident vectors.

6. How to use this calculator

This calculator will help you find the triangle area formed by coincident vectors.
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 area of the triangle formed by these coincident vectors.
$\bold{Solution:1}$
$\bold{Step \space 1:}$ We will apply the determinant method to find the cross product.
$\bold{Step \space 2:}$ $\bold{\vec{a} \space x \space \vec{b} =\begin{vmatrix} \^{i} & \^{j} & \^{k} \\ 2 & 1 & 3 \\ 4 & 2 & 1 \end{vmatrix}}$ = $\bold{\^{i}\begin{vmatrix} 1 & 3 \\ 2 & 1 \end{vmatrix}} - \bold{\^{j}\begin{vmatrix} 2 & 4 \\ 3 & 1 \end{vmatrix}} + \bold{\^{k}\begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} = -5i + 10j + 0k = -5i + 10j }$
$\bold{Step \space 3:}$ So the value of $\bold{\frac{1}{2}|\vec{a} \space x \space \vec{b}|}$ = $\bold{\frac{1}{2}\sqrt{(-5)^2 + (10)^2}}$ = $\frac{5\sqrt{5}}{2}$

$\bold{Question:2}$
Given $\bold{\vec{a}}$ = 3i - 1j + 2k, $\bold{\vec{b}}$ = 2i + 4j - 1k find the area of the triangle formed by these coincident vectors.
$\bold{Solution:2}$
$\bold{Step \space 1:}$ We will apply the determinant method to find the cross product.
$\bold{Step \space 2:}$ $\bold{\vec{b} \space x \space \vec{a} =\begin{vmatrix} \^{i} & \^{j} & \^{k} \\ 3 & -1 & 2 \\ 2 & 4 & -1 \end{vmatrix}}$ = $\bold{\^{i}\begin{vmatrix} -1 & 2 \\ 4 & -1 \end{vmatrix}} - \bold{\^{j}\begin{vmatrix} 3 & 2 \\ 2 & -1 \end{vmatrix}} + \bold{\^{k}\begin{vmatrix} 3 & -1 \\ 2 & 4 \end{vmatrix} = -7i + 7j + 14k }$
$\bold{Step \space 3:}$ So the value of $\bold{\frac{1}{2}|\vec{b} \space x \space \vec{a}|}$ = $\bold{\frac{1}{2}.\sqrt{(-7)^2 + (7)^2 + (14)^2}}$ = $\frac{7\sqrt{6}}{2}$

8. Frequently Asked Questions (FAQs)

Can the area of a triangle formed by coincident vectors be negative?

No, the area is always non-negative.

What happens if the vectors are not coincident?

The cross product yields a vector perpendicular to the plane of the triangle, and its magnitude represents twice the area.

Is the order of vectors important in the formula?

No, the order matters only in the cross-product but not in its magnitude.

Can the area be zero?

Yes, if the vectors are coincident, resulting in a zero cross-product.

Can this method be extended to find the area of any triangle?

Yes, it is a general method applicable to any triangle.

9. What are the real-life applications?

In physics and engineering, understanding the area of triangles formed by coincident vectors is vital for calculating surface area moments and determining equilibrium in structures.

10. Conclusion

As we conclude our exploration into the realm of triangles formed by coincident vectors, the elegance of mathematical principles becomes apparent. The ability to calculate the area with precision not only enriches our understanding of vectors but also finds valuable applications in various scientific and engineering domains. The geometric beauty of vector mathematics continues to unfold, offering insights and tools for navigating the complexities of spatial relationships.

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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