Notes
Questions
For numbers 1-5, use \(\vec{u}=2\vec{i}-3\vec{j}\) and \(\vec{v}=4\vec{i}+2\vec{j}\)
\(\textbf{1)}\) Find the dot product \(\vec{u}\cdot\vec{v}\)
The dot product \(\vec{u}\cdot\vec{v}=2\)
\(\textbf{2)}\) Find the magnitude of \(\vec{v}\)
The magnitude is \(|\vec{v}|=2\sqrt{5}\)
\(\textbf{3)}\) Find the unit vector in the same direction as \(\vec{v}\)
The unit vector is \(\displaystyle\frac{2\sqrt{5}}{5} \vec{i}+\displaystyle\frac{\sqrt{5}}{5} \vec{j}\)
\(\textbf{4)}\) Find the scalar projection of \(\vec{u}\) onto \(\vec{v}\).
The scalar projection is \(\displaystyle\frac{\sqrt{5}}{5}\)
\(\textbf{5)}\) Find the vector projection of \(\vec{u}\) onto \(\vec{v}\).
The vector projection is \(\displaystyle\frac{2}{5}\vec{i}+\displaystyle\frac{1}{5}\vec{j}\)
For numbers 6-10 use \(\vec{r}=2\vec{i}+5\vec{j}-1\vec{k}\) and \(\vec{s}=3\vec{i}-4\vec{j}+6\vec{k}\)
\(\textbf{6)}\) Find the dot product \(\vec{r}\cdot\vec{s}\)
The dot product is \(\vec{r}\cdot\vec{s}=-20\)
\(\textbf{7)}\) Find the magnitude of \(\vec{s}\)
The magnitude is \(\sqrt{51}\)
\(\textbf{8)}\) Find the unit vector in the same direction as \(\vec{s}\)
The unit vector is \(\displaystyle\frac{3\sqrt{51}}{51}\vec{i}-\displaystyle\frac{4\sqrt{51}}{51}\vec{j}+\displaystyle\frac{6\sqrt{51}}{51}\vec{k}\)
\(\textbf{9)}\) Find the scalar projection of \(\vec{r}\) onto \(\vec{s}\).
The scalar projection is \(-\displaystyle\frac{20\sqrt{51}}{51}\)
\(\textbf{10)}\) Find the vector projection of \(\vec{r}\) onto \(\vec{s}\).
The vector projection is \(-\displaystyle\frac{60}{51}\vec{i}+\displaystyle\frac{80}{51}\vec{j}-\displaystyle\frac{120}{51}\vec{k}\)
See Related Pages\(\)
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