So I'm trying to imagine the dot product intuitively and I'm wondering if it would be right to think of the dot product, if we're using two vectors A and B (imagine that there's an arrow over those letters), as the multiplied amount vector A is in the B direction or vice versa since it's the same value. like when I try to do the dot product I align one of the vectors on an axis and project the other vector onto the axis-aligned vector and the length of that is the amount the projected vector is in the direction of the axis-aligned vector. I don't know how else to think of it. and if I'm imagining it right then how does this intuitively relate the law of cosines?
sorry I don't know that language most of you use to make what I'm trying to explain more clean
The dot product can be intuitively understood as the area of a rectangle with one side one of the vectors and the other side the orthogonal projection of the other vector onto that vector. Or it can be understood as half of the difference between the areas of squares formed by two vectors and the area of the square formed by their difference. The former tells us that the dot product of a vector with itself is the area of a square formed by that vector, which leads to the latter understanding, and the latter understanding is equivalent to the law of cosines.
special thank you
so my understanding was wrong? and how does dot product have anything to do with the area of a rectangle? I'm having a hard time imagining this; I understood the projection onto the vector part but that creates a right triangle that's used to kind of create another vector super imposed onto the vector being projected; like there's vector A and then vector B is projected onto it and the spot that the projection hits A is the length Bcos(theta) and you times the magnitude of vector A by Bcos(theta) to get ABcos(theta), or it can be done vice versa, no? like doesn't this mean that the dot product is just the magnitude of the line that's formed when you times the magnitude of one full vector by the length of the projection of the other vector onto that vector? I'm not disagreeing, of course, since I don't understand this shit one bit, but I just need to shatter my current view of the dot product first
No, you're understanding appears to be correct. You just didn't mention that you multiply the magnitude of the projection by the magnitude of the vector it's projected onto. Multiplying the length of two lines gives you the area of the rectangle formed by them. So from this we first derive that
A·A = |A|^2
If we have two vectors A and B with angle θ between them then we can form a triangle by drawing a vector C from the end of A to the end of B.
C = A-B
C·C = (A-B)·(A-B)
C·C = A·A - A·B - B·A + B·B
|C|^2 = |A|^2 - 2 A·B + |B|^2 <- law of cosines
A·B = ( |A|^2 + |B|^2 - |C|^2 )/2
this is quite an ideological post anon. Have you tried blaming capitilism?
ok I guess I'm just really dense but:
magnitude of the projection? the line perpendicular to the vector being projected onto? (or does the projection refer to the length of the line ontop of the vector being projected onto; the amount the other vector is in the direction of the vector being projected onto?) If it's the first thing, then how do I find the length of the projection; the line perpendicular to the vector being projected onto? I lost you at the A•A = lAl^2 part, basically, and I understand the rest of it until the law of cosines part (the second to last step); doesn't the law of cosines have a cos(theta) in it, and what does the very last part have to do with anything else?
The projection of one vector onto another is the "amount" or "magnitude of the former that is going in the direction of the latter. Suppose we have two vectors A and B. The the dot product is the the magnitude of one of the vectors, say A, multiplied by the magnitude/amount of B in the direction of the A (aka projection of B onto A). This is why the dot product of some vector with itself is the square of its magnitude; its the same vector so of course all of its magnitude is directed in the same direction as itself.
thank you that was crystal clear, which kind of answered the whole thread, for the most part;
so what was the rectangle bit? that guy was referring to the line creating the right angle times the magnitude of the vector touching the right angle which equates to the area of a rectangle with the vector length and projection width? I'm just trying to intuitively connect my understanding of the dot product with the rectangle thing and the law of cosines now -- which doesn't really matter but I have like an obsessive compulsion I can't move on until I understand everything