Scalar and Vector Projections

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Notes

Notes for Scalar Projection

Notes for Vector Projection

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 \(\vec{u}\cdot\vec{v}\)
Show Dot Product
The dot product \(\vec{u}\cdot\vec{v}=2\)

\(\textbf{2)}\) Find the magnitude of \(\vec{v}\)
Show Magnitude
The magnitude is \(|\vec{v}|=2\sqrt{5}\)

\(\textbf{3)}\) Find the unit vector in the same direction as \(\vec{v}\)
Show Unit Vector
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 \(\vec{v}\).
Show Scalar Projection
The scalar projection is \(\displaystyle\frac{\sqrt{5}}{5}\)

\(\textbf{5)}\) Find the vector projection of \(\vec{u}\) onto \(\vec{v}\).
Show Vector Projection
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 \(\vec{r}\cdot\vec{s}\)
Show Dot Product
The dot product is \(\vec{r}\cdot\vec{s}=-20\)

\(\textbf{7)}\) Find the magnitude of \(\vec{s}\)
Show Magnitude
The magnitude is \(\sqrt{51}\)

\(\textbf{8)}\) Find the unit vector in the same direction as \(\vec{s}\)
Show Unit Vector
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 \(\vec{s}\).
Show Scalar Projection
The scalar projection is \(-\displaystyle\frac{20\sqrt{51}}{51}\)

\(\textbf{10)}\) Find the vector projection of \(\vec{r}\) onto \(\vec{s}\).
Show Vector Projection
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\(\)

\(\bullet\text{ Displacement Vectors}\)
\(\,\,\,\,\,\,\,\,(x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}…\)

\(\bullet\text{ Magnitude, Direction, and Unit Vectors}\)
\(\,\,\,\,\,\,\,\,|\vec{u}|=\sqrt{a^2+b^2}…\)

\(\bullet\text{ Dot Product}\)
\(\,\,\,\,\,\,\,\,a \cdot b=x_1 x_2+ y_1 y_2…\)

\(\bullet\text{ Parallel and Perpendicular Vectors}\)
\(\,\,\,\,\,\,\,\,⟨8,2⟩ \text{ and } ⟨−4,−1⟩…\)

\(\bullet\text{ Scalar and Vector Projections}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{a \cdot b}{|b|^2} \, \vec{b}…\)

\(\bullet\text{ Cross Product}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Equation of a Plane}\)
\(\,\,\,\,\,\,\,\,Ax+By+Cz=D…\)

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