Triangle law of vector addition and the Pythagoras theorem

[akashpandey]

TL;DR

Triangle law of vector addition and pythagoras theorem

i know its pretty basic but please give some insight for
triangle law of vector addition and pythgoras theorem.
because ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem

 

[jbriggs444]

Summary: Triangle law of vector addition and pythagoras theorem

i know its pretty basic but please give some insight for
triangle law of vector addition and pythgoras theorem.
because ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem

If by "triangle law", you mean the law of cosines, check out what happens when the angle is 90 degrees.

 

[fresh_42]

Summary: Triangle law of vector addition and pythagoras theorem

i know its pretty basic but please give some insight for
triangle law of vector addition and pythgoras theorem.
because ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem

"The triangular law" for vector addition is what happens if we add to vectors to a third vector. All have direction and length. Pythagoras is a special case of the cosine theorem. It needs a right angle, otherwise we need the cosine theorem. And even in case of a right angle, it only says something about the length of the resulting vector, nothing about direction. So "triangular law" is what happens, and Pythagoras how to calculate the length.

 

[akashpandey]

Suppose there is a vector a of magnitude 5 units to the east, another vector b of magnitude 6 units to the north. To find magnitude of vector a + vector b,
By the triangle law of vector addition, it is 5 + 6 units = 11 units.
By Pythagorean theorem, it is sqrt(5^2 + 6^2) = sqrt(61)
Which answer is right? If so, why is the other wrong?
Thank you!

 

[jbriggs444]

By the triangle law of vector addition, it is 5 + 6 units = 11 units.
By Pythagorean theorem, it is sqrt(5^2 + 6^2) = sqrt(61)
Which answer is right? If so, why is the other wrong?

Pythagoras is correct. There is no such thing as the "triangle law" you have quoted.

 

[vanhees71]

Perhaps the OP has the cosine theorem in mind?

Take three points A, B, C in Euclidean space. Then you have
$$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.$$
Thus
$$|BC|^2=\overrightarrow{BC}^2=(\overrightarrow{AC}-\overrightarrow{AB})^2 = |AC|^2 +|AB|^2 -2 |AC| |AB| \cos(\angle{BAC}).$$
If the angle is $\pi/2$ the cosine term vanishes, and you have Pythagoras's Theorem as a special case.