Simple qn on vectors and scalars

[gunblaze]

ok...
Is it possible to use the formula to decide whther a physical quantity is a scalar or a vector?

eg:
Velocity=change in displacement/time

Thus, since displacement is a vector,
therefore, velocity is also a vector>?

And for density...
Density=mass/volume

SInce mass and volume are both scalar,
Therefore, density is a scalar>?

 

[Davorak]

ok...
Is it possible to use the formula to decide whther a physical quantity is a scalar or a vector?

eg:
Velocity=change in displacement/time

Thus, since displacement is a vector,
therefore, velocity is also a vector>?

And for density...
Density=mass/volume

SInce mass and volume are both scalar,
Therefore, density is a scalar>?

Yes and yes. If the input to a formula is a vectors or vectors then the answer to the formula will be a vector. Unless that is the formula has a dot product, or absolute value. Or in some cases it is possible to take one component of a vector and use that as a scalar answer.

 

[dextercioby]

With 2 vectors in 3D multiple operations can be made and the result can be
1.A scalar.
2.A pseudovector.
3.A vector.
4.A second rank tensor...

Daniel.

 

[cepheid]

Hey Daniel, that's interesting. So, I'm assuming that you arrive at 1 using the dot product, and 3 using the cross product. But what is a pseudovector? How do you make one? And is a second-rank tensor just a matrix? Again, how do you combine the two vectors to make one? Thanks.

I hope the answer will not be too hard for me to understand...

 

[dextercioby]

Not really.At a vector you reach by vector-space operations...Addition and scalar multiplication.Cross-product is a special case of exterior product and the result is a pseudovector...As for second rank tensor,well it's the tensor/dyadic (ancient name) product...Scalar product (inner product) is just a contracted tensor product.

Daniel.

 

[gunblaze]

But... how about work done, potential energy and kinetic energy?

eg:
Work done=force*distance
work done=mass *acceleration*distance

Acceleration is a vector but that doesn't mean work done is a vector too rite?

 

[dextercioby]

"distance" is not a scalar...It is the "displacement vector"...In the definition of work,there is a dot/scalar product which makes the result a scalar...

Daniel.

 

[Crosson]

Gunblaze, the answer to your question have to do with vector multiplication, which I am guessing you have not been exposed to.

F = ma (bold quantities are vectors)

This equation is a scalar times a vector. The reason they are called scalars is that they will scale a into a different vector F

Now look at work:

[tex]W = \vec{F} \cdot \vec{d}[\tex]

This is a multiplication of two vectors, fundamentally different than Newtons Second law. Notice that the answer is a number (a scalar) , so this is a scalar product between vectors. Another example:

[tex]KE = \frac{1}{2}m(\vec{v})^2 = \frac{1}{2}m\ vec{v} \cdot \vec{v}[\tex]

So kinetic energy, again a scalar, is a product of v with itself.

 

[kdawg090]

How do you graphically add vectors? For example Vector A is equal to a force of .980 N at a compass angle of 35 degrees and Vector B has a force of 1.96 N at 165 degrees.

 

[dextercioby]

If u've given not only modulus,but also angles,then those angles are defined wrt certain axis of coordinates.Then u'll simply add their components...

Daniel.

 

[ramollari]

How do you graphically add vectors? For example Vector A is equal to a force of .980 N at a compass angle of 35 degrees and Vector B has a force of 1.96 N at 165 degrees.

Well, the strategy for 2D is this. Choose a coordinate system, with one of the axes preferably along one of the vectors. Find the angles that the vecotrs make with the axes, and break into x and y components. Find the moduli of the components by multiplying with $$sin$$ or $$cos$$ and give (+) or (-) signs to the components, then add them as scalars to get the net x and y components.
To get the modulus of the resulting vector use the Pythagorean theorem, while to get the angle use the $$arctan$$.

 

[Dracovich]

I'm curious as to when you use each method between 2 vectors. I've just been following equations up until now, but i would like to understand why, and for what reason (and to what end) i use the cross product between vectors, and when the dot. In what situations does one use each method??

 

[dextercioby]

Depends of what he wishes to write.Usually,quantities are DEFINED using these matematical operations (with vectors),and therefore your last question loses meaning.


Daniel.