Pythagorean Theorem proof Vector Calc. Due tomorrow 9am. PLEASE HELP

In summary, the Pythagorean theorem states that if a, b, and c are vectors in Rn and a + b = c and a.b = 0, then ||a||2 + ||b||2 = ||c||2. This theorem is named after the ancient Greek mathematician Pythagoras and is used to find the length of the sides in a right triangle. The hint given suggests that the dot product of a and b being 0 means they are perpendicular, creating a right angle in the triangle. This leads to the use of the cosine rule in proving the Pythagorean theorem.
  • #1
rgalvan2
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Pythagorean Theorem proof Vector Calc. Due tomorrow 9am. PLEASE HELP!

Prove the Pythagorean theorem. That is, if a,b, and c are vectors in Rn such that a + b = c and a.b = 0, then ||a||2 + ||b||2 = ||c||2. Why is this called the Pythagorean theorem.

Hint given: Given the hypotheses, you have a right triangle. Why?

Help please! This is due tomorrow morning and I cannot figure it out.

||x||= length of variable
a.b= dot product
 
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  • #2


rgalvan2 said:
Hint given: Given the hypotheses, you have a right triangle. Why?

Well what does [tex] a \cdot b = 0 [/tex] tell you about the vectors a and b?
 
  • #3


Draw a triangle,explain why the angle is 90 degrees and then think cosine rule.
 

1. What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical principle that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

2. How is the Pythagorean Theorem used in vector calculus?

In vector calculus, the Pythagorean Theorem can be used to find the magnitude or length of a vector. By breaking down a vector into its component parts and applying the theorem, we can find the length of the vector in a given direction.

3. What is the proof of the Pythagorean Theorem?

The most commonly known proof of the Pythagorean Theorem involves using squares to represent the areas of the sides of a right triangle. By rearranging the squares in a specific way, it can be shown that the theorem holds true for any right triangle.

4. How do you apply the Pythagorean Theorem in vector calculus problems?

In vector calculus problems, the Pythagorean Theorem can be used to find the magnitude or length of a vector, as well as to solve for unknown sides or angles in a triangle. It is also used in finding the distance between points in multi-dimensional space.

5. Are there any other proofs of the Pythagorean Theorem?

Yes, there are several different proofs of the Pythagorean Theorem, including geometric proofs, algebraic proofs, and even proofs using calculus. Each proof offers a different perspective on the theorem and helps to deepen our understanding of its fundamental principles.

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