d90575…: “Like the fine icosahedron”

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Let $\vec {u} = \begin {bmatrix} 1\\3\end {bmatrix}$ , $\vec {v} = \begin {bmatrix} -5\\2\end {bmatrix}$ , $\vec {w} = \begin {bmatrix} 1\\0\\-1\end {bmatrix}$ , $\vec {x} = \begin {bmatrix} 2\\3\\1\end {bmatrix}$ .
Compute the following:
$\vec {u} \cdot \vec {u} = \answer {10}$
$||\vec {u}|| = \sqrt {\answer {10}}$
$||\vec {u}||^2 = \answer {10}$
$\vec {u} \cdot \vec {v} = \answer {1}$
$\vec {x}\cdot \vec {w} = \answer {1}$
$||\vec {v}||^2 = \answer {29}$
$||\vec {w} - \vec {x}|| = \sqrt {\answer {14}}$

Is the computation $\vec {u} \cdot \vec {w}$ possible?

Yes No

Let $\vec {u} = \begin {bmatrix} -1\\7\\2\end {bmatrix}$ . Compute the following:

$\left (\frac {1}{\vec {u}\cdot \vec {u}}\right )\vec {u} = \begin {bmatrix} \frac {\answer {-1}}{\answer {54}} \\ \frac {\answer {7}}{\answer {54}} \\\frac {\answer {1}}{\answer {27}}\end {bmatrix}$

What is the distance between $\vec {x} = \begin {bmatrix} 3\\4\end {bmatrix}$ and $\vec {y} = \begin {bmatrix} 5\\-5 \end {bmatrix}$ ?

85 $\sqrt {85}$ 65 $\sqrt {65}$

True/False: Two vectors in $\RR ^n$ are orthogonal if and only if their inner product (dot product) is zero.

True False

True/False: The vectors $\vec {x} = \begin {bmatrix} -1\\4\\3\end {bmatrix}$ and $\vec {y} = \begin {bmatrix} 2\\6\\2 \end {bmatrix}$ are orthogonal.

True False

True/False: $\vec {v} = \begin {bmatrix} 1/2\\1/2\end {bmatrix}$ is a unit vector.

True False

Find a unit vector in the direction of $\vec {v}=\begin {bmatrix} -1\\2\\-2\\4\end {bmatrix}$ .

First find the length of $\vec {v}$ : $||\vec {v}|| = \answer {5}$ .

Then find a unit vector $\vec {u} = \begin {bmatrix} -\answer {1}/\answer {5} \\ \answer {2}/\answer {5}\\ -\answer {2}/\answer {5} \\ \answer {4}/\answer {5}\end {bmatrix}$ .

Find a unit vector in the direction of $\vec {v}=\begin {bmatrix} 1\\1\end {bmatrix}$ .

First find the length of $\vec {v}$ : $||\vec {v}|| = \sqrt {\answer {2}}$ .

Then find a unit vector $\vec {u} = \begin {bmatrix} 1/\sqrt {\answer {2}} \\ 1/\sqrt {\answer {2}}\\ \end {bmatrix}$ .