Displacement & Pythagorean Theorem: Triangle ABC

[lrl4565]

Alright, we have triangle abc with hypotenuse c. So, if you add vector a and vector b, the answer is vector c.

Now, according to the pythagorean theorem, this would not make sense. But the pythagorean theorem is DISTANCE. I am guessing that this phenomenon has something to do with using displacement?

 

[Filip Larsen]

Note, that the pythagorean theorem corresponds to the particular geometric situation where vector a and b are orthogonal on each other and in this case, the pythogorean theorem relates the length of the three vectors. For general vectors a and b, you need the law of cosine that includes the angle between a and b in order to related the length of a and b with the length of c.

 

[lrl4565]

My book says vector a + vector b = vector c. It uses (x,y) coordinates.

Vector a: (x1,y1)
Vector b: (x2,y2)
Vector c: (x1+x2, y1+y2)

Filip Larsen, what you just described gives me the magnitude of the displacement.

Am I measuring distance or displacement? Is the vector all about displacement?

 

[jtbell]

Displacement is only one example of a vector quantity. It's the first one that most physics textbooks introduce. There are many others: velocity, acceleration, force, momentum, angular momentum, electric field, magnetic field, ...

In your example, the total distance traveled would be the sum of the lengths (magnitudes) of the displacement vectors a and b:

$$d_{total} = \sqrt{x_1^2 + y_1^2} + \sqrt{x_2^2 + y_2^2}$$

The total displacement would be simply the vector c, which has magnitude

$$\sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2}$$

and a direction (angle) given by

$$\tan \theta = \frac{y_1 + y_2}{x_1 + x_2}$$

 

[Naty1]

You can perhaps get some additional insights here:

http://en.wikipedia.org/wiki/Vector_addition#Vector_addition_and_subtraction

See the the diagram with red,green and blue i,j,k description...

 

[lrl4565]

Aren't we talking about displacement vectors?

http://en.wikipedia.org/wiki/Displacement_vectorHow do they relate to Euclidean vectors? Is that the link to other types of vectors?

 

[diazona]

What do you mean by "Euclidean vectors"?

 

[lrl4565]

Naty1's link is for Euclidean Vectors

 

[diazona]

Oh, silly me should've clicked on that.

From what I read in the article, it seems like a displacement vector is just one example of a Euclidean vector. Although it kind of depends on how you define "Euclidean vector" - there's a bit of ambiguity in the article, probably because different groups of people (e.g. physicists vs. mathematicians) have different definitions for the term.