91Ó°ÊÓ

Determine whether the set of vectors in \(R^{n}\) is orthogonal, orthonormal, or neither. $$\\{(2,-4),(2,1)\\}$$

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

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

Find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle,\) (b) \(\|\mathbf{u}\|,(\mathrm{c})\|\mathbf{v}\|,\) and (d) \(d(\mathbf{u}, \mathbf{v})\) for the given inner product defined in \(R^{n}\). \(\mathbf{u}=(3,4), \quad \mathbf{v}=(5,-12), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\)

Determine whether the sets are orthogonal. $$S_{1}=\operatorname{span}\left\\{\left[\begin{array}{r} 2 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]\right\\} \quad S_{2}=\operatorname{span}\left\\{\left[\begin{array}{r} -1 \\ 2 \\ 0 \end{array}\right]\right\\}$$

Determine whether the set of vectors in \(R^{n}\) is orthogonal, orthonormal, or neither. $$\\{(3,-2),(-4,-6)\\}$$

Find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle,\) (b) \(\|\mathbf{u}\|,(\mathrm{c})\|\mathbf{v}\|,\) and (d) \(d(\mathbf{u}, \mathbf{v})\) for the given inner product defined in \(R^{n}\). \(\mathbf{u}=(1,1), \quad \mathbf{v}=(7,9), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\)

Find the length of the vector. \(\mathbf{v}=(0,1)\)

Determine whether the sets are orthogonal. $$S_{1}=\operatorname{span}\left\\{\left[\begin{array}{r} -3 \\ 0 \\ 1 \end{array}\right]\right\\} \quad S_{2}=\operatorname{span}\left\\{\left[\begin{array}{l} 2 \\ 1 \\ 6 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right]\right\\}$$

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