91Ó°ÊÓ

Prove: For any complex numbers \(z, w \in \mathbf{C},(\text { i) } \overline{z+w}=\bar{z}+\bar{w}\) (ii) \(\overline{z w}=\bar{z} \bar{w}\) (iii) \(\bar{z}=z\)

Let \(u=(1,-2,4), v=(3,5,1), w=(2,1,-3) .\) Find: (a) \(3 u-2 v\) (b) \(5 u+3 v-4 w\) \(\begin{array}{llll}\text { (c) } u \cdot v, & u \cdot w, & v \cdot w ;\end{array}\) (d) \(\|u\|,\|v\|\) (e) \(\cos \theta,\) where \(\theta\) is the angle between \(u\) and \(v\) \((\mathrm{f}) \quad d(u, v)\) \((g) \quad \operatorname{proj}(u, v)\)

Find \(k\) so that \(u\) and \(v\) are orthogonal, where: (a) \(u=(3, k,-2), v=(6,-4,-3)\) (b) \(u=(5, k,-4,2), v=(1,-3,2,2 k)\) (c) \(u=(1,7, k+2,-2), v=(3, k,-3, k)\)

Consider a moving body \(B\) whose position at time \(t\) is given by \(R(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+3 t \mathbf{k} .\) [Then \(V(t)=d R(t) / d t \text { and } A(t)=d V(t) / d t \text { denote, respectively, the velocity and acceleration of } B .]\) When \(t=1,\) find for the body \(B:\) (a) position; (b) velocity \(v\) (c) speed \(s\) (d) acceleration \(a\)

Prove the following properties of the cross product: (a) \(u \times v=-(v \times u)\) (d) \(u \times(v+w)=(u \times v)+(u \times w)\) (b) \(u \times u=0\) for any vector \(u\) (e) \((v+w) \times u=(v \times u)+(w \times u)\) (c) \((k u) \times v=k(u \times v)=u \times(k v)\) \((\mathrm{f}) d(u \times v) \times w=(u \cdot w) v-(v \cdot w) u\)

Prove that the norm in \(\mathbf{C}^{n}\) satisfies the following laws: \(\left[\mathrm{N}_{1}\right]\) For any vector \(u,\|u\| \geq 0 ;\) and \(\|u\|=0\) if and only if \(u=0\) \(\left[\mathrm{N}_{2}\right]\) For any vector \(u\) and complex number \(z,\|z u\|=| z\|u\|\) \(\left[\mathrm{N}_{3}\right]\) For any vectors \(u\) and \(v,\|u+v\| \leq\|u\|+\|v\|\)