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
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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