Chapter 5 - Inner Product Spaces - Review Exercises - Page 284: 2

(a) $\| u \| = \sqrt{5}$ (b) $\| v \| = \sqrt{13}$ (c) $\langle u, v \rangle =4$ (d) $d(u,v)= \sqrt{10}.$

Work Step by Step

Let $u=(-1,2), \quad v=(2,3)$, then we have (a) $\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{u_1^2+u_2^2}=\sqrt{5}$ (b) $\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{v_1^2+v_2^2}=\sqrt{13}$ (c) $\langle u, v \rangle =u_1v_1+u_1v_2=4$ (d) $d(u,v)=\| u-v \|=\| (-3,-1) \|=\sqrt{10}.$

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