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MATH115 GEOMETRY AND CALCULUS

Workshop Exercises 1

1.

Let $\mathbf{u}=(1\;\;0\;\;1)$ , $\mathbf{v}=(2\;\;2\;\;1)$ and $\mathbf{w}=(-1\;\;2\;\;-1)$ .

i) Compute $|\mathbf{u}+|\mathbf{v}|\mathbf{w}|$ .

ii) Compute $(\mathbf{u}\cdot\mathbf{v})\mathbf{w}$ .

iii) Find a unit vector (i.e., a vector of length 1) which is orthogonal to both $\mathbf{u}$ and $\mathbf{v}$ .
2.

Let ${\bf u}$ , ${\bf v}$ be vectors in ${\mathbb{R}}^{n}$ .

i) Show that if ${\bf v}\cdot{\bf v}=0$ then ${\bf v}={\bf 0}$ .

ii) Show that $|{\bf u}|^{2}+|{\bf v}|^{2}\geq 2|{\bf u}\cdot{\bf v}|$ .

iii) Show that $|{\bf u}+{\bf v}|^{2}+|{\bf u}-{\bf v}|^{2}=2(|{\bf u}|^{2}+|{\bf v}|^{2})$ .
3.

True or false?

i) There exist vectors ${\bf u}$ and ${\bf v}$ in ${\mathbb{R}}^{2}$ such that ${\bf u}\cdot{\bf v}=|{\bf u}|^{2}+|{\bf v}|^{2}$ .

ii) If ${\bf u}\cdot{\bf v}={\bf u}\cdot{\bf w}$ then ${\bf v}={\bf w}$ .
4.

In each of the following cases, find $\cos\theta$ , where $\theta$ is the angle between ${\bf u}$ and ${\bf v}$ .

i) ${\bf u}=(1\;\;-1)$ , ${\bf v}=(-2\;\;5)$ ,

ii) ${\bf u}=(2\;\;0\;\;-1)$ , ${\bf v}=(-1\;\;3\;\;2)$ ,

iii) ${\bf u}=(-3\;\;1\;\;-1\;\;5)$ , ${\bf v}=(-4\;\;0\;\;8\;\;1)$ .
5.

i) Find the equation of the line in ${\mathbb{R}}^{2}$ which passes through the points $(2,-2)$ and $(7,-1)$ .

ii) Find the equation of the plane in ${\mathbb{R}}^{3}$ with normal vector $(1\;\;-2\;\;3)$ and which passes through the point $(2,3,-1)$ .

iii) Find an equation describing the line in ${\mathbb{R}}^{3}$ passing through the points $(-2,3,5)$ and $(1,0,-1)$ .