Took me way too long to find a decent video on how the projection vector formula is derived. Thank you very much for explaining it in such a clear and concise manner!
Have used this to great affect with a digital compass I’m making at home. For those who aren’t aware, earths magnetic field is actually 3 dimensional. The sensor may not be aligned perfectly to the “world coordinates”……but I can get the vector of magnetic field measurement, then dot product against a vertical unit vector and remove from the signal, so I’m left with horizontal measurements. For product against east and north unit vectors, gives me magnitudes in each of those directions. Then I can calculate magnetic north direction and display as an angle on the little LCD screen. This stuff does have practical applications in the real world. Love it
And before someone says about cross product (lol) I do realise that. And have used it too…..but was conveying this part of my story as it relates to this video
Loved your last couple of videos on dot product. When I taught this I always started with the definition as : (1) u.v = |u| |v| cos(t) rather than the (2) u1v1 + u2v2 which I saw a a consequence of the definition. I know that (2) follows the rules of definitions but here is why I chose (1) as a more basic logical starting place. (for me anyway) There are several concepts in physics (which my senior students were already familiar with) that require multiplying 2 vector quantities to arrive at a meaningful scalar. The most common example in high school physics is that the work done by an unbalanced force is given by (3) W = |the component of F in the direction of the displacement| * |dispalcement|. Assuming F and D have the same initial point and separated by an angle t, you need to drop a perpendicular from the tip of F to vector D. You now have 2 vertical componenets of F - one in the direction of D and the other perpendicular to D (which does no work on the moving mass).This turns (3) into W = |F|cos(t)|*|D| = |F| |D| cos(t) which then gets defined as "dot product of F and D". Notice that you can drop the perpendicular from the tip of D to meet F and proceed in exactly the same way. If t is acute we get a positive work which then increases the speed (KE) in the direction of D. If t is obtuse we think of negative work as slowing the speed or increasing it in the opposite direction to D. I then did exactly what you did with the cosine law to arrive at the componenet form, u1v1 + u2v2, of the dot product imposed on a co-orinate system. This form, as you pointed out, gives us a more convenient/faster way of finding the angle between 2 vectors. Now that we have the "scalar projection of u on v", (magnitude) we now multiply that by a unit vector in the direction of v, and out pops the "vector projection of u on v" formula as v.v = |v|^2 I really like your approach and as always - keep up the good work. So much fun.
I also used to verify the following with my students. Not needed but nice to see it actually works!: u = , v = , p = Your dotted line vector is u - p = so that p "dot" (u - p) = -232/81 + 152/81 + 80/81 = 0 Verifying that p and (u-p) are indeed perpendicular.
Yeah, it doesn't seem right to me either. Without doing any calculations, the magnitude has to be at least the absolute value of the largest coordinate, which is 8/9 in this case.
Kindly explain me why the inventor mathematician defined a dot product as a scalar where as the cross product as a vector which is perpendicular to the both vectors and it's detection is depend on the right hand finger rule.
dude , can you do a video on inner product next and what exactly does it do ? i've heard it's similar to dot product and it kinda makes sense with summation being similar to integration but I just don't get it at all . thanks ☺
For 2 vectors -->u and -->v, the dot product calculates: (magnitude of the componenet of of -->u in the direction of -->v) multiplied by the (magnitude of -->v) . That is, | -->u|*|-->v|*cos(t). Very important scalar in physics. For example: W (scalar) = -->F "dot" -->d .
Took me way too long to find a decent video on how the projection vector formula is derived. Thank you very much for explaining it in such a clear and concise manner!
This is one of the best videos on vector projection I’ve watched. Excellent introduction with step-by-step explanations. Thank you for sharing!
Thanks so much! I'm glad it helped.
Have used this to great affect with a digital compass I’m making at home.
For those who aren’t aware, earths magnetic field is actually 3 dimensional. The sensor may not be aligned perfectly to the “world coordinates”……but I can get the vector of magnetic field measurement, then dot product against a vertical unit vector and remove from the signal, so I’m left with horizontal measurements.
For product against east and north unit vectors, gives me magnitudes in each of those directions.
Then I can calculate magnetic north direction and display as an angle on the little LCD screen.
This stuff does have practical applications in the real world. Love it
And before someone says about cross product (lol) I do realise that. And have used it too…..but was conveying this part of my story as it relates to this video
Loved your last couple of videos on dot product. When I taught this I always started with the definition as : (1) u.v = |u| |v| cos(t) rather than the (2) u1v1 + u2v2 which I saw a a consequence of the definition. I know that (2) follows the rules of definitions but here is why I chose (1) as a more basic logical starting place. (for me anyway)
There are several concepts in physics (which my senior students were already familiar with) that require multiplying 2 vector quantities to arrive at a meaningful scalar. The most common example in high school physics is that the work done by an unbalanced force is given by
(3) W = |the component of F in the direction of the displacement| * |dispalcement|.
Assuming F and D have the same initial point and separated by an angle t, you need to drop a perpendicular from the tip of F to vector D. You now have 2 vertical componenets of F - one in the direction of D and the other perpendicular to D (which does no work on the moving mass).This turns (3) into W = |F|cos(t)|*|D| = |F| |D| cos(t) which then gets defined as "dot product of F and D". Notice that you can drop the perpendicular from the tip of D to meet F and proceed in exactly the same way. If t is acute we get a positive work which then increases the speed (KE) in the direction of D. If t is obtuse we think of negative work as slowing the speed or increasing it in the opposite direction to D.
I then did exactly what you did with the cosine law to arrive at the componenet form, u1v1 + u2v2, of the dot product imposed on a co-orinate system. This form, as you pointed out, gives us a more convenient/faster way of finding the angle between 2 vectors.
Now that we have the "scalar projection of u on v", (magnitude) we now multiply that by a unit vector in the direction of v, and out pops the "vector projection of u on v" formula as v.v = |v|^2
I really like your approach and as always - keep up the good work. So much fun.
oh my god, such a good video, thank you!
I also used to verify the following with my students. Not needed but nice to see it actually works!: u = , v = , p =
Your dotted line vector is u - p =
so that p "dot" (u - p) = -232/81 + 152/81 + 80/81 = 0 Verifying that p and (u-p) are indeed perpendicular.
As somebody who loves geometric algebra, I've to say this: all you need is to notice (proj u)v=(proj u).v=u.v, so proj u=u.v/v.
It's ok to use u•v/v here .However, In general multivectors don't commute, so you should write (u•v)(v**-1)
@@appybane8481 Thanks. I just agree with the book of Alan Macdonald to always mean post-multiply with the inverse when using the division notation.
Umm not sure about the magnitude part at the end. Wouldn't the 4/9 need to be multiplied by the magnitude of v?
Yeah, it doesn't seem right to me either. Without doing any calculations, the magnitude has to be at least the absolute value of the largest coordinate, which is 8/9 in this case.
It’s wrong in fact. The magnitude of the projection is u*v divided by the magnitude of v. It’s 4/3
The magnitude is actually 12/9
@simoncaprello...correct
Thank you for pointing that out! I just trimmed out that part of the video!
Kindly explain me why the inventor mathematician defined a dot product as a scalar where as the cross product as a vector which is perpendicular to the both vectors and it's detection is depend on the right hand finger rule.
See my long comment above. Your answer might be there.
dude , can you do a video on inner product next and what exactly does it do ? i've heard it's similar to dot product and it kinda makes sense with summation being similar to integration but I just don't get it at all . thanks ☺
As I remember, dot product is a particular kind of inner product. Inner products can be done on functions
For 2 vectors -->u and -->v, the dot product calculates: (magnitude of the componenet of of -->u in the direction of -->v) multiplied by the (magnitude of -->v) .
That is, | -->u|*|-->v|*cos(t). Very important scalar in physics. For example: W (scalar) = -->F "dot" -->d .
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First!