dot product, mag cross product, and cross product of vectors ...
Author
hello, my question is basically on the main differences between these, i have some idea, but sorta need a clear explanation and / or confirmation on what i know ... since what i find when i searh are long and drawn out confusing examples.
so the dot product of 2 vectors equals a scalar?
Scalar = |V1| * |V2| * Cos (theta)
does this mean he dot product gives me the magnitude to the resultant?
now the magnitude cross product ...
|V1| * |V2| * Sin (theta)
what does this give me? if the dot produvt gives me the resultant?
and finally the cross product?
just V1 * V2 .. im utterly confused on this one,
what's really the point other then finding the resultant?
are these 3 methods all just shorthand ways to finding the resultant instead of breaking the vectors down in components and using traditional trigonometry?
thanks for any help or clear links on explanations explaining these in advance.
Maybe you should read something about geometric algebra.
There was also a thread around some time ago.
There was also a thread around some time ago.
Dot product returns a scaler
remember that |V1| is the magnitude of vector V1
One potential use of this is that (V1 dot V2)/(|V1|*|V2|) = cos(theta) and with inverse trig you can find the angle between two vectors, potentially a very useful fact.
Cross product returns another vector with the magnitude you gave. The direction of the vector is perpendicular to the two crossed vectors. In other words
V3 = V1 x V2
V3 is perpendicular to both V1 and V2. This of course means that V3 can point in two possible directions, you can use a right hand rule to determine which.
This is exceedingly useful because it provides an easy way to find the normal to a given surface w/o calculus. The normal is useful both for graphics and for modeling physical situations. Also some formulas require the cross product - F=q*v x B comes to mind.
You could use traditional trig, but dot and cross products are easier to read and to understand. The dot product also has a lot of significance in abstract linear algebra in which case it goes beyond simple trig. However for our purposes I suppose you could just think of them as shorthand.
I hope this helps and that I don't have any glaring errors
remember that |V1| is the magnitude of vector V1
One potential use of this is that (V1 dot V2)/(|V1|*|V2|) = cos(theta) and with inverse trig you can find the angle between two vectors, potentially a very useful fact.
Cross product returns another vector with the magnitude you gave. The direction of the vector is perpendicular to the two crossed vectors. In other words
V3 = V1 x V2
V3 is perpendicular to both V1 and V2. This of course means that V3 can point in two possible directions, you can use a right hand rule to determine which.
This is exceedingly useful because it provides an easy way to find the normal to a given surface w/o calculus. The normal is useful both for graphics and for modeling physical situations. Also some formulas require the cross product - F=q*v x B comes to mind.
You could use traditional trig, but dot and cross products are easier to read and to understand. The dot product also has a lot of significance in abstract linear algebra in which case it goes beyond simple trig. However for our purposes I suppose you could just think of them as shorthand.
I hope this helps and that I don't have any glaring errors
Not sure what you mean by the resultant. The resultant is just the result of any operation on 1 or more operands, so it's nothing in particular as you seem to think it is. That said I'll give you a quick rundown of the dot and cross products
Dot Product:
To find the dot product (which is actually the Euclidean inner product) you component-wise multiply the 2 vectors together and then add the components of the result. In example, the dot product of V1=(x1,y1,z1) and V2=(x2,y2,z2) is
scalar = x1*x2 + y1*y2 + z1*z2
which is also equal to
scalar = |V1| * |V2| * cos(theta)
as you noted yourself. This is very useful when you need to find the angle between 2 vectors as well as for determining when 2 vectors are othogonal (ie theta=0), in which case dot(V1,V2)=0. It is also used when finding the projection of 1 vector onto another.
Cross Product:
The cross-product only applies to 3 dimensions. (Actually, that's not entirely true, but cross products in higher dimensions take more operands and don't apply to 3D rendering, so I'll pretend that it is true). Given the same 2 vectors as above, the cross product is calculated as follows:
cross = (y1*z2-z1*y2, z1*x2-x1*z2, x1*y2-x2*y1)
and as you noted, the magnitude of the result is equal to
|cross| = |V1| * |V2| * sin(theta)
What the gives cross product gives you is a little more tangible than the dot product. The cross product of 2 vectors gives you a 3rd vector that is perpendicular to the first 2. For example, if you were to cross a unit vector along the +ve X axis with a unit vector along the +ve Y axis the result would be a unit vector along the +ve Z axis. Geometrically, if you were to build a parallelogram with V1 and V2 again, the magnitude of the cross product is equal to the area of the parallelogram.
If you have any more questions just ask. Hope that helps!!
Dot Product:
To find the dot product (which is actually the Euclidean inner product) you component-wise multiply the 2 vectors together and then add the components of the result. In example, the dot product of V1=(x1,y1,z1) and V2=(x2,y2,z2) is
scalar = x1*x2 + y1*y2 + z1*z2
which is also equal to
scalar = |V1| * |V2| * cos(theta)
as you noted yourself. This is very useful when you need to find the angle between 2 vectors as well as for determining when 2 vectors are othogonal (ie theta=0), in which case dot(V1,V2)=0. It is also used when finding the projection of 1 vector onto another.
Cross Product:
The cross-product only applies to 3 dimensions. (Actually, that's not entirely true, but cross products in higher dimensions take more operands and don't apply to 3D rendering, so I'll pretend that it is true). Given the same 2 vectors as above, the cross product is calculated as follows:
cross = (y1*z2-z1*y2, z1*x2-x1*z2, x1*y2-x2*y1)
and as you noted, the magnitude of the result is equal to
|cross| = |V1| * |V2| * sin(theta)
What the gives cross product gives you is a little more tangible than the dot product. The cross product of 2 vectors gives you a 3rd vector that is perpendicular to the first 2. For example, if you were to cross a unit vector along the +ve X axis with a unit vector along the +ve Y axis the result would be a unit vector along the +ve Z axis. Geometrically, if you were to build a parallelogram with V1 and V2 again, the magnitude of the cross product is equal to the area of the parallelogram.
If you have any more questions just ask. Hope that helps!!
Author
ok i think i understand now, so basically to get the dot product of two vectors you can apply the same method as you would to add them ... so if V1 and V2 are 2 dimensional they could be broken down into (i'll use "a" instead of writing theta) V1x, V1y, V1a, and V2x, V2y, and V2a this would mean to multiply them traditionally i would use.
V3x = |V1x| * |V2x|
V3y = |V1y| * |V2y|
then to find the magnitude of V3 i could use pythagorean theorm (sure can't spell that).
but i could use dot product to find this same answer? just by knowing the magnitudes of the two vetors and the angle between them? so if i knew the angle between V1 and V2 i could use (phi being the angle between the vectors)
V3 = |V1| * |V2| * COS(phi)
this should give me the same answer if i knew the angles from the horizontal or vertical and broke the vectors into components then multiplied like above.
but then for the cross product, i get another vector, but this vector is perpendicular to the direction of the vector formed when two vectors are multiplyed together?
and by resultant (that is what my physics teachers have told me to call it) i am refering to the vector that is fromed when you add them together, but i guess that word is vauge and i dont know why they would tell me to call it that.
anyways ... i think i understand now what dot product and cross products are, but if my above ramblings are wrong please tell me :) ... thanks
V3x = |V1x| * |V2x|
V3y = |V1y| * |V2y|
then to find the magnitude of V3 i could use pythagorean theorm (sure can't spell that).
but i could use dot product to find this same answer? just by knowing the magnitudes of the two vetors and the angle between them? so if i knew the angle between V1 and V2 i could use (phi being the angle between the vectors)
V3 = |V1| * |V2| * COS(phi)
this should give me the same answer if i knew the angles from the horizontal or vertical and broke the vectors into components then multiplied like above.
but then for the cross product, i get another vector, but this vector is perpendicular to the direction of the vector formed when two vectors are multiplyed together?
and by resultant (that is what my physics teachers have told me to call it) i am refering to the vector that is fromed when you add them together, but i guess that word is vauge and i dont know why they would tell me to call it that.
anyways ... i think i understand now what dot product and cross products are, but if my above ramblings are wrong please tell me :) ... thanks
You cannot multiply vectors together, you can add them, and do scalar multiplication, and do dot products and cross products, thats it.
the dot product is a scalar, no components.
V1*V2 = V1x*V2x+V1y*V2y
it is not related to any vector V3
The cross product does not return a vector perpendicular to the product of v1 and v2 because the product of v1 and v2 is meaningless. It is perpendicular to both vectors. It is only defined in three space. It doesn't make sense in 2 space (nothing is perpendicular to 2 linearly independent vectors) and in 4 space is not unique (lots of vectors are perpendicular to 2 linearly independent vectors)
the dot product is a scalar, no components.
V1*V2 = V1x*V2x+V1y*V2y
it is not related to any vector V3
The cross product does not return a vector perpendicular to the product of v1 and v2 because the product of v1 and v2 is meaningless. It is perpendicular to both vectors. It is only defined in three space. It doesn't make sense in 2 space (nothing is perpendicular to 2 linearly independent vectors) and in 4 space is not unique (lots of vectors are perpendicular to 2 linearly independent vectors)
This topic is closed to new replies.
Advertisement
Popular Topics
Advertisement