Proving Inequality: Vector Method for Cos2A+Cos2B+Cos2C in Triangle ABC

Thread starter utkarshakash
Start date Aug 16, 2013
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Inequality

In summary, the Vector Method for Proving Inequality in Triangle ABC is a mathematical approach that uses vectors to prove the inequality Cos2A+Cos2B+Cos2C ≥ 1 in any triangle ABC. It works by representing the cosine values as dot products of vectors and using geometric properties to manipulate the inequalities. The benefits of using this method include a unique and visual approach, deeper understanding of geometric properties, and applicability to various other inequalities. However, it may be complex and limited to specific types of inequalities. In real-world situations, it can be applied in fields such as engineering, construction, and physics to ensure stability and analyze forces and motion in geometric shapes.

Aug 16, 2013

utkarshakash

Gold Member

Homework Statement

In a triangle ABC, prove by vector method cos2A+cos2B+cos2C≥ -3/2.

Homework Equations

The Attempt at a Solution

I can change the LHS of the inequality to the form
(cos2A i + cos2B j + cos2C k).(i+j+k)

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Aug 16, 2013

tiny-tim

Science Advisor

Homework Helper

hi utkarshakash!

hint: 2A + 2B + 2C = 360° …

so draw a circle!

FAQ: Proving Inequality: Vector Method for Cos2A+Cos2B+Cos2C in Triangle ABC

What is the Vector Method for Proving Inequality in Triangle ABC?

The Vector Method for Proving Inequality in Triangle ABC is a mathematical approach that uses vectors to prove the inequality Cos2A+Cos2B+Cos2C ≥ 1 in any triangle ABC. It involves representing the cosine values as dot products of vectors and using geometric properties to manipulate the inequalities.

How does the Vector Method work?

The Vector Method works by representing the cosine values of angles A, B, and C in triangle ABC as dot products of vectors. By using geometric properties, such as the triangle inequality and the dot product properties, we can manipulate the inequality to prove that Cos2A+Cos2B+Cos2C ≥ 1.

What are the benefits of using the Vector Method?

The Vector Method offers a unique and visual approach to proving inequalities in triangles. It allows for a deeper understanding of the geometric properties involved and can be used to prove various other inequalities in different geometric shapes.

What are the limitations of the Vector Method?

The Vector Method can be complex and may require a good understanding of vector operations and geometric properties. It may also be limited to proving specific types of inequalities and may not be applicable in all cases.

How can the Vector Method be applied in real-world situations?

The Vector Method can be applied in real-world situations where inequalities need to be proven in geometric shapes. For example, it can be used in engineering and construction to ensure the stability and strength of structures, or in physics to analyze forces and motion in different shapes.