Notes

 

\(\text{Unit Vector}=\displaystyle\frac{\vec{v}}{|\vec{v}|}= \)\(\left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|} \right\rangle \)

 

 

\(\text{Magnitude of 2d Vector}\)
\(\text{If } \vec{v}=\langle a,b \rangle, \text{then }|\vec{v}|=\sqrt{a^2+b^2}\)

 

 

\(\text{Magnitude of 3d Vector}\)
\(\text{If } \vec{v}=\langle a,b,c \rangle, \text{then }|\vec{v}|=\sqrt{a^2+b^2+c^2}\)

 

 

\(\text{Vector Notation}\)
\(\langle a,b \rangle \text{means the same thing as }a\vec{i}+b\vec{j}\)
\(\langle a,b,c \rangle \text{means the same thing as }a\vec{i}+b\vec{j}+c\vec{k}\)

 

 

Problems, Solutions, & Videos

\(\textbf{1)}\) Find the unit vector in the same direction as \(\vec{a}=3\vec{i}-4\vec{j}\).

Show Unit Vector

\(\,\,\,\text{The unit vector is }\frac{3}{5}\vec{i}- \frac{4}{5}\vec{j} \)

Show Work

\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = \sqrt{3^2+4^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = \sqrt{9+16}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = \sqrt{25}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = 5\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{a}}{|\vec{a}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{3\vec{i}-4\vec{j}}{5}\)
\(\,\,\,\text{The unit vector is } \frac{3}{5}\vec{i}- \frac{4}{5}\vec{j} \)

 

\(\textbf{2)}\) Find the unit vector in the same direction as \(\vec{b}=\langle 3,-4,6 \rangle\).

Show Unit Vector

\(\,\,\,\text{The unit vector is } \left\langle \frac{3}{\sqrt{61}},-\frac{4}{\sqrt{61}},\frac{6}{\sqrt{61}}\right\rangle\,\,\, \text{ or }\,\,\,\left\langle\frac{3\sqrt{61}}{61},-\frac{4\sqrt{61}}{61},\frac{6\sqrt{61}}{61}\right\rangle\)

Show Work

\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{b}| = \sqrt{3^2+4^2+6^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{b}| = \sqrt{9+16+36}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{b}| = \sqrt{61}\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{b}}{|\vec{b}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\left\langle\frac{3,-4,6}{\sqrt{61}}\right\rangle\)
>\(\,\,\,\text{The unit vector is } \left\langle \frac{3}{\sqrt{61}},-\frac{4}{\sqrt{61}},\frac{6}{\sqrt{61}}\right\rangle\,\,\, \text{ or }\,\,\,\left\langle\frac{3\sqrt{61}}{61},-\frac{4\sqrt{61}}{61},\frac{6\sqrt{61}}{61}\right\rangle\)

 

\(\textbf{3)}\) Find the unit vector in the same direction as \(\vec{v}=3\vec{i}+4\vec{j}+12\vec{k}\).

Show Unit Vector

\(\,\,\,\text{The unit vector is } \frac{3}{13} \vec{i}+\frac{4}{13}\vec{j}+\frac{12}{13}\vec{k}\)

Show Work

\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = \sqrt{3^2+4^2+12^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = \sqrt{9+16+144}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = \sqrt{169}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = 13\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{v}}{|\vec{v}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{3\vec{i}+4\vec{j}+12\vec{k}}{13}\)
\(\,\,\,\text{The unit vector is } \frac{3}{13} \vec{i}+\frac{4}{13}\vec{j}+\frac{12}{13}\vec{k}\)

 

\(\textbf{4)} \) Find the unit vector in the same direction as \(\vec{n}=\langle 2,-7 \rangle\).

Show Unit Vector

\(\,\,\,\text{The unit vector is } \left\langle \frac{2}{\sqrt{53}},\frac{4}{\sqrt{53}} \right\rangle\)

Show Work

\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{n}| = \sqrt{2^2+(-7)^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{n}| = \sqrt{4+49}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{n}| = \sqrt{53}\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{n}}{|\vec{n}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\langle 2,-7 \rangle}{\sqrt{53}}\)
\(\,\,\,\text{The unit vector is } \left\langle \frac{2}{\sqrt{53}},\frac{-7}{\sqrt{53}} \right\rangle\)

 

 

See Related Pages\(\)

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

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

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

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

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

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

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