Find the distance between the vectors \(\mathrm{u}\) and \(\mathrm{v}\) where i) \(\mathrm{u}=(1,7), \mathrm{v}=(6,-5) ;\) ii) $\mathrm{u}=(3,-5,4), \mathrm{v}=(6,2,-1)$ iii) \(\mathrm{u}=(5,3,-2,-4,1) \cdot \mathrm{v}=(2,-1,0,-7,2)\)

Let $\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \mathrm{x}_{\mathrm{n}}\right),\left(\mathrm{y}_{1}, \mathrm{y}_{2}, \ldots, \mathrm{y}_{\mathrm{n}}\right),\left(\mathrm{z}_{1}, \mathrm{z}_{2}, \ldots \ldots \mathrm{z}_{\mathrm{n}}\right)$ be three vectors in \(\mathrm{R}^{\mathrm{n}}\). Verify the following properties of the dot product using these vectors : a) \(\mathrm{X} \cdot \mathrm{X} \geq 0 ; \mathrm{X} \cdot \mathrm{X}=0\) if and only if \(\mathrm{X}=0\) b) \(X \cdot Y=Y \cdot X\) c) $(\mathrm{X}+\mathrm{Y}) \cdot \mathrm{Z}=\mathrm{X} \cdot \mathrm{Y}+\mathrm{X} \cdot \mathrm{Z}$ d) $(\mathrm{CX}) \cdot \mathrm{Y}=\mathrm{X} \cdot(\mathrm{cY})=\mathrm{c}(\mathrm{X} \cdot \mathrm{Y})$

Find the area of the triangle determined by the points $\mathrm{P}_{1}(2,2,0), \mathrm{P}_{2}(-1,0,1)\( and \)\mathrm{P}_{3}(0,4,3)$ by using the cross-product.

Let \(\mathrm{u}=(2,-1,3)\) and \(\mathrm{v}=(4,-1,2)\) be vectors in \(\mathrm{R}^{3}\). Find the orthogonal projection of \(\mathrm{u}\) on \(\mathrm{v}\) and the component of \(\mathrm{u}\) orthogonal to v.

Use the Gram-Schmidt process to transform \([(1,0,1),(1,2,-2),(2,-1,1)]\) into an orthogonal basis for \(\mathrm{R}^{3}\). Assume the standard inner product.

Let \(\mathrm{u}=(4,0,1,2,0), \mathrm{v}=(2,1,-1,1,1) .\) Find: a) $\mathrm{u} \cdot \mathrm{v} ; \mathrm{b})\|\mathrm{u}\|,\|\mathrm{v}\| ; \mathrm{c}\( ) the projection of \)\mathrm{u}\( onto \)\mathrm{v}$ and the projection of u orthogonal to \(\mathrm{v}\).