I am independently studying Ap Physics C: Mechanics and I am learning about dot product. However, I don’t understand how Scalar = A x B = |A| x BII. (Where A and B are vectors, the || represent magnitude, the BII means the component of B parallel to A) I can’t seem to be able to rationalize it and it frustrates me.
Edit: A bit of clarification added
In: 3
Try thinking of it in terms of a concrete example – the standard choice is the amount of work done by an object being displaced with displacement S in a field of force F.
The work done – a scalar, in Joules – is equal to the displacement multiplied by the force in the same direction as that displacement.
So you take the absolute value of the displacement (|S|) and you multiply it by the force parallel to that displacement (FII)
You can safely discard both vectors directional information because you’ve already removed any components that aren’t in the relevant direction, and have reduced the problem to a one-dimensional one.
If you’re in a physics class you’re probably thinking about vectors as objects with magnitude and direction. Vectors are actually more complicated than that. They are actually mathematical structures. A “vector space” is a set of objects that has certain qualities and a “vector” is one element from a vector space.
A vector space actually only has vector + vector addition and vector X scalar multiplication defined on it. So it’s not actually necessarily defined what it means to multiply two vectors. So we get to make stuff up.
In physics we generally use the vector space of R^(3) (The points that make up 3D space) we usually come up with two vector X vector multiplication operations. One that results in a scalar (dot product) and one that results in a vector (cross product).
When we use these spacial vectors, we usually define the dot product to be mean “how similar are these two vectors”. So if two vectors point in exactly the same direction the dot product is just the product of the magnitudes of the vectors. When they point in opposite directions we get the negative of the magnitudes. When you multiply two vectors that are perpendicular to each other you get 0. Anywhere in between these extremes you get what you put above. It should be noted that the dot product is communicative so:
A*B=|A| X BII = |B| X AII
However this is purely conventional. This is what a dot product is because that’s what we said it is. It’s useful to define this operation for spacial vectors. It tells you how inline two vectors are, which is sometimes important to know.
When you get deeper into physics and your vectors spaces get weirder your dot product will mean different things. It’ll mean whatever is useful at the time. For instance you’ll later find out that the set of all continuous functions is a vector space, so continuous functions are actually vectors. A dot product on those involves multiplying the functions together and doing some calculus on them. In that case it tells us how much “overlap” these functions have.
So the question shouldn’t be why does the dot product do this, it should be why did we define an operation that does this.
From your title, it seems you think that Bll is a vector, so you don’t understand how Vector*Scalar gives Scalar.
If so, than that is incorrect. The dot(A, B) = |A|*|Bll| – only lengths are multiplied.
However, it might be still difficult to understand, why many properties of dot product are true. Here is some things, that might help:
1. You can move vectors around. For studying Dot, it is nice to have vectors tail from the same point.
2. Two vectors always have a common 2D plane. Which means that studying only a 2D case is enough.
3. Note, that |Bll|=|B|*cos (angle_between(A,B)). Which means, Dot(A, B) = |A|*|B|*cos (angleAB). From this, it is obvious, that Dot(A,B)=Dot(B,A)
4. To prove that Dot(C, A+B)=Dot(C,A)+Dot(C,B), notice that (A+B)ll=All +Bll.
5. To prove Dot(A, k*B)=k*Dot(A,B), notice that |kB|=k*|B|.
6. To derive a coordinate formula, remember that A=a_x***X** + a_y***Y** + a_z***Z**, where a_x,a_y,a_z are coordinates of A, and **X,Y,Z** are the basis vectors.
7. Expand Dot(a_x***X** + a_y***Y** + a_z***Z**, b_x***X** + b_y***Y** + b_z***Z**) with rules 4 and 5 to get a coordinate formula.
8. When X,Y,Z is ORTHONORMAL (all basis vectors are length 1, and all perpendicular to each other), the formula in 7 gets much simpler: Dot(X,X)=Dot(Y,Y)=Dot(Z,Z)=1, while all other Dot(…, …)=0. So in this case, Dot(A,B)=a_x*b_x + a_y*b_y + a_z*b_z. Note, that this is a **special case** when the basis is orthonormal – if it is not, use formula from 7.
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