| r → − r → ′ | 2 = r 2 + r ′ 2 − 2 r ⋅ r ′ cos ⁡ ( θ )How is this possible?I could understand why, if we had brackets instead of abs values. But as it is I cannot understand how to operate with the square of an abs. value

Question

|              r      →        −                    r        →              ′              |        2    =      r    2    +      r          ′      2        −  2  r  ⋅      r    ′    cos  ⁡  (  θ  )How is this possible?I could understand why, if we had brackets instead of abs values. But as it is I cannot understand how to operate with the square of an abs. value

Zara Pratt · Accepted Answer

Since             r      →        and                     r        →              ′   are elements of a vector space, and not elements of an integral domain,       |                  r        →              −                            r          →                    ′        |   does not denote absolute value, but the norm (magnitude) of the vector             r      →        −                    r        →              ′    ,, although it actually would be more correct to denote it as       |          |                        r          →                    −                                    r            →                          ′            |        |  . By definition,       |          |      v      |        |    =      v    ∙          v        ,, where   ∙ denotes the inner product (the dot product, in this case). So             |              |                              r            →                          −                                            r              →                                ′                |            |        2    =      (                  r        →              −                            r          →                    ′        )    ∙            (                        r          →                    −                                    r            →                          ′            )        =            r      →        ∙                    r        →              −            r      →        ∙                              r          →                    ′        −                    r        →              ′    ∙                    r        →              +                    r        →              ′    ∙                              r          →                    ′        =            |              |                              r            →                          |            |        2    +            |              |                                            r              →                                ′                |            |        2    −  2            r      →        ∙                              r          →                    ′        cos  ⁡  (  θ  )  .This, is actually, the law of cosines in disguise.

|vec(r)-vec(r)'|^2=r^2+r'^(2)-2r * r' cos(theta) How is this possible? I could understand why, if we had brackets instead of abs values. But as it is I cannot understand how to operate with the square of an abs. value

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