Chapter 5 - Inner Product Spaces - 5.1 Length and Dot Product in Rn - 5.1 Exercises - Page 236: 82

Using the property of the dot product, we have \begin{align*} \frac{1}{4}\|{u}+{v}\|^{2}-\frac{1}{4}\|{u}-{v}\|^{2}&=\frac{1}{4}(u+v)\cdot (u+v)-\frac{1}{4}(u-v)\cdot (u-v)\\ &=\frac{1}{4}(u\cdot u+2u\cdot v+v\cdot v)\\ &-\frac{1}{4}(u\cdot u-2u\cdot v+v\cdot v)\\ &=-frac{1}{4}(4 u\cdot v)\\ &=u\cdot v. \end{align*}

Using the property of the dot product, we have \begin{align*} \frac{1}{4}\|{u}+{v}\|^{2}-\frac{1}{4}\|{u}-{v}\|^{2}&=\frac{1}{4}(u+v)\cdot (u+v)-\frac{1}{4}(u-v)\cdot (u-v)\\ &=\frac{1}{4}(u\cdot u+2u\cdot v+v\cdot v)\\ &-\frac{1}{4}(u\cdot u-2u\cdot v+v\cdot v)\\ &=-frac{1}{4}(4 u\cdot v)\\ &=u\cdot v. \end{align*}