91Ó°ÊÓ

(a) determine whether the set of vectors in \(R^{n}\) is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for \(R^{n}\). \(\left\\{\left(\frac{3}{5}, \frac{4}{5}\right),\left(-\frac{4}{5}, \frac{3}{5}\right)\right\\}\)

determine whether the points are collinear. If so, find the line \(y=c_{B}+c_{I} x\) that fits the points. $$ (-2,0),(0,2),(2,2) $$

Find the length of the vector. $$\mathbf{v}=(5,-3,-4)$$

Find the cross product of the unit vectors [ where \(\mathbf{i}=(1,0,0)\) \(\mathbf{j}=(\mathbf{0}, \mathbf{1}, \mathbf{0}), \text { and } \mathbf{k}=(\mathbf{0}, \mathbf{0}, \mathbf{1})] .\) Sketch your result. $$\mathbf{i} \times \mathbf{k}$$

Find the cross product of the unit vectors [ where \(\mathbf{i}=(1,0,0)\) \(\mathbf{j}=(\mathbf{0}, \mathbf{1}, \mathbf{0}), \text { and } \mathbf{k}=(\mathbf{0}, \mathbf{0}, \mathbf{1})] .\) Sketch your result. $$\mathbf{k} \times \mathbf{j}$$

(a) determine whether the set of vectors in \(R^{n}\) is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for \(R^{n}\). \(\left\\{(2,1),\left(\frac{1}{3},-\frac{2}{3}\right)\right\\}\)

Find the length of the vector. $$\mathbf{v}=(2,0,-5,5)$$

Show that the function defines an inner product on \(R^{2},\) where \(\mathbf{u}=\left(u_{1}, u_{2}\right)\) and \(\mathbf{v}=\left(v_{1}, v_{2}\right)\) $$\langle\mathbf{u}, \mathbf{v}\rangle= 2 u_{1} v_{2}+u_{2} v_{1}+u_{1} v_{2}+2 u_{2} v_{2}$$

Find (a) \(\|\mathbf{u}\|,(\mathbf{b})\|\mathbf{v}\|,\) and \((\mathbf{c})\|\mathbf{u}+\mathbf{v}\|\) $$\mathbf{u}=\left(-1, \frac{1}{4}\right), \quad \mathbf{v}=\left(4,-\frac{1}{8}\right)$$

(a) determine whether the set of vectors in \(R^{n}\) is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for \(R^{n}\). \(\\{(4,-1,1),(-1,0,4),(-4,-17,-1)\\}\)