A unit vector is a vector with magnitude \(1\) that points in the same direction as a given vector. To find one, first calculate the magnitude of the original vector, then divide each component by that magnitude. These problems include unit vectors in two dimensions, three dimensions, and both component notation and \(\vec{i},\vec{j},\vec{k}\) notation.
Notes
\(\text{Unit Vector}=\displaystyle\frac{\vec{v}}{|\vec{v}|}= \)\(\left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|} \right\rangle \)
\(\text{Magnitude of 2d Vector}\)
\(\text{If } \vec{v}=\langle a,b \rangle, \text{then }|\vec{v}|=\sqrt{a^2+b^2}\)
\(\text{Magnitude of 3d Vector}\)
\(\text{If } \vec{v}=\langle a,b,c \rangle, \text{then }|\vec{v}|=\sqrt{a^2+b^2+c^2}\)
\(\text{Vector Notation}\)
\(\langle a,b \rangle \text{means the same thing as }a\vec{i}+b\vec{j}\)
\(\langle a,b,c \rangle \text{means the same thing as }a\vec{i}+b\vec{j}+c\vec{k}\)
Practice Problems
\(\textbf{1)}\) Find the unit vector in the same direction as \(\vec{a}=3\vec{i}-4\vec{j}\).
\(\,\,\,\text{The unit vector is }\frac{3}{5}\vec{i}- \frac{4}{5}\vec{j} \)
\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = \sqrt{3^2+(-4)^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = \sqrt{9+16}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = \sqrt{25}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{a}| = 5\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{a}}{|\vec{a}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{3\vec{i}-4\vec{j}}{5}\)
\(\,\,\,\text{The unit vector is } \frac{3}{5}\vec{i}- \frac{4}{5}\vec{j} \)
\(\textbf{2)}\) Find the unit vector in the same direction as \(\vec{b}=\langle 3,-4,6 \rangle\).
\(\,\,\,\text{The unit vector is } \left\langle \frac{3}{\sqrt{61}},-\frac{4}{\sqrt{61}},\frac{6}{\sqrt{61}}\right\rangle\,\,\, \text{ or }\,\,\,\left\langle\frac{3\sqrt{61}}{61},-\frac{4\sqrt{61}}{61},\frac{6\sqrt{61}}{61}\right\rangle\)
\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{b}| = \sqrt{3^2+(-4)^2+6^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{b}| = \sqrt{9+16+36}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{b}| = \sqrt{61}\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{b}}{|\vec{b}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\langle3,-4,6\rangle}{\sqrt{61}}\)
\(\,\,\,\text{The unit vector is } \left\langle \frac{3}{\sqrt{61}},-\frac{4}{\sqrt{61}},\frac{6}{\sqrt{61}}\right\rangle\,\,\, \text{ or }\,\,\,\left\langle\frac{3\sqrt{61}}{61},-\frac{4\sqrt{61}}{61},\frac{6\sqrt{61}}{61}\right\rangle\)
\(\textbf{3)}\) Find the unit vector in the same direction as \(\vec{v}=3\vec{i}+4\vec{j}+12\vec{k}\).
\(\,\,\,\text{The unit vector is } \frac{3}{13} \vec{i}+\frac{4}{13}\vec{j}+\frac{12}{13}\vec{k}\)
\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = \sqrt{3^2+4^2+12^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = \sqrt{9+16+144}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = \sqrt{169}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{v}| = 13\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{v}}{|\vec{v}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{3\vec{i}+4\vec{j}+12\vec{k}}{13}\)
\(\,\,\,\text{The unit vector is } \frac{3}{13} \vec{i}+\frac{4}{13}\vec{j}+\frac{12}{13}\vec{k}\)
\(\textbf{4)} \) Find the unit vector in the same direction as \(\vec{n}=\langle 2,-7 \rangle\).
\(\,\,\,\text{The unit vector is } \left\langle \frac{2}{\sqrt{53}},\frac{-7}{\sqrt{53}} \right\rangle\)
\(\,\,\,1^{\text{st}} \text{ step, find the magnitude.}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{n}| = \sqrt{2^2+(-7)^2}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{n}| = \sqrt{4+49}\)
\(\,\,\,\,\,\,\,\,\,\displaystyle |\vec{n}| = \sqrt{53}\)
\(\,\,\,2^{\text{nd}} \text{ step, find the unit vector.}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\vec{n}}{|\vec{n}|}\)
\(\,\,\,\,\,\,\,\,\,\text{Unit vector is } \displaystyle\frac{\langle 2,-7 \rangle}{\sqrt{53}}\)
\(\,\,\,\text{The unit vector is } \left\langle \frac{2}{\sqrt{53}},\frac{-7}{\sqrt{53}} \right\rangle\)
\(\textbf{5)}\) Find the unit vector in the same direction as \(\vec{u}=\langle 5,12\rangle\).
\(\text{The unit vector is } \left\langle\frac{5}{13},\frac{12}{13}\right\rangle\)
\(|\vec{u}|=\sqrt{5^2+12^2}\)
\(|\vec{u}|=\sqrt{25+144}\)
\(|\vec{u}|=13\)
\(\text{Unit vector is }\displaystyle\frac{\vec{u}}{|\vec{u}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle5,12\rangle}{13}\)
\(\text{The unit vector is } \left\langle\frac{5}{13},\frac{12}{13}\right\rangle\)
\(\textbf{6)}\) Find the unit vector in the same direction as \(\vec{w}=\langle -6,8\rangle\).
\(\text{The unit vector is } \left\langle-\frac{3}{5},\frac{4}{5}\right\rangle\)
\(|\vec{w}|=\sqrt{(-6)^2+8^2}\)
\(|\vec{w}|=\sqrt{36+64}\)
\(|\vec{w}|=10\)
\(\text{Unit vector is }\displaystyle\frac{\vec{w}}{|\vec{w}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle-6,8\rangle}{10}\)
\(\text{The unit vector is } \left\langle-\frac{3}{5},\frac{4}{5}\right\rangle\)
\(\textbf{7)}\) Find the unit vector in the same direction as \(\vec{p}=\langle 2,-3,6\rangle\).
\(\text{The unit vector is } \left\langle\frac{2}{7},-\frac{3}{7},\frac{6}{7}\right\rangle\)
\(|\vec{p}|=\sqrt{2^2+(-3)^2+6^2}\)
\(|\vec{p}|=\sqrt{4+9+36}\)
\(|\vec{p}|=7\)
\(\text{Unit vector is }\displaystyle\frac{\vec{p}}{|\vec{p}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle2,-3,6\rangle}{7}\)
\(\text{The unit vector is } \left\langle\frac{2}{7},-\frac{3}{7},\frac{6}{7}\right\rangle\)
\(\textbf{8)}\) Find the unit vector in the same direction as \(\vec{r}=8\vec{i}-15\vec{j}\).
\(\text{The unit vector is } \frac{8}{17}\vec{i}-\frac{15}{17}\vec{j}\)
\(|\vec{r}|=\sqrt{8^2+(-15)^2}\)
\(|\vec{r}|=\sqrt{64+225}\)
\(|\vec{r}|=17\)
\(\text{Unit vector is }\displaystyle\frac{\vec{r}}{|\vec{r}|}\)
\(\text{Unit vector is }\displaystyle\frac{8\vec{i}-15\vec{j}}{17}\)
\(\text{The unit vector is } \frac{8}{17}\vec{i}-\frac{15}{17}\vec{j}\)
\(\textbf{9)}\) Find the unit vector in the same direction as \(\vec{m}=-7\vec{i}+24\vec{j}\).
\(\text{The unit vector is } -\frac{7}{25}\vec{i}+\frac{24}{25}\vec{j}\)
\(|\vec{m}|=\sqrt{(-7)^2+24^2}\)
\(|\vec{m}|=\sqrt{49+576}\)
\(|\vec{m}|=25\)
\(\text{Unit vector is }\displaystyle\frac{\vec{m}}{|\vec{m}|}\)
\(\text{Unit vector is }\displaystyle\frac{-7\vec{i}+24\vec{j}}{25}\)
\(\text{The unit vector is } -\frac{7}{25}\vec{i}+\frac{24}{25}\vec{j}\)
\(\textbf{10)}\) Find the unit vector in the same direction as \(\vec{q}=\langle -9,-40\rangle\).
\(\text{The unit vector is } \left\langle-\frac{9}{41},-\frac{40}{41}\right\rangle\)
\(|\vec{q}|=\sqrt{(-9)^2+(-40)^2}\)
\(|\vec{q}|=\sqrt{81+1600}\)
\(|\vec{q}|=41\)
\(\text{Unit vector is }\displaystyle\frac{\vec{q}}{|\vec{q}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle-9,-40\rangle}{41}\)
\(\text{The unit vector is } \left\langle-\frac{9}{41},-\frac{40}{41}\right\rangle\)
\(\textbf{11)}\) Find the unit vector in the same direction as \(\vec{t}=\langle 1,2,2\rangle\).
\(\text{The unit vector is } \left\langle\frac{1}{3},\frac{2}{3},\frac{2}{3}\right\rangle\)
\(|\vec{t}|=\sqrt{1^2+2^2+2^2}\)
\(|\vec{t}|=\sqrt{1+4+4}\)
\(|\vec{t}|=3\)
\(\text{Unit vector is }\displaystyle\frac{\vec{t}}{|\vec{t}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle1,2,2\rangle}{3}\)
\(\text{The unit vector is } \left\langle\frac{1}{3},\frac{2}{3},\frac{2}{3}\right\rangle\)
\(\textbf{12)}\) Find the unit vector in the same direction as \(\vec{d}=\langle 4,0,-3\rangle\).
\(\text{The unit vector is } \left\langle\frac{4}{5},0,-\frac{3}{5}\right\rangle\)
\(|\vec{d}|=\sqrt{4^2+0^2+(-3)^2}\)
\(|\vec{d}|=\sqrt{16+0+9}\)
\(|\vec{d}|=5\)
\(\text{Unit vector is }\displaystyle\frac{\vec{d}}{|\vec{d}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle4,0,-3\rangle}{5}\)
\(\text{The unit vector is } \left\langle\frac{4}{5},0,-\frac{3}{5}\right\rangle\)
\(\textbf{13)}\) Find the unit vector in the same direction as \(\vec{x}=6\vec{i}+2\vec{j}+3\vec{k}\).
\(\text{The unit vector is } \frac{6}{7}\vec{i}+\frac{2}{7}\vec{j}+\frac{3}{7}\vec{k}\)
\(|\vec{x}|=\sqrt{6^2+2^2+3^2}\)
\(|\vec{x}|=\sqrt{36+4+9}\)
\(|\vec{x}|=7\)
\(\text{Unit vector is }\displaystyle\frac{\vec{x}}{|\vec{x}|}\)
\(\text{Unit vector is }\displaystyle\frac{6\vec{i}+2\vec{j}+3\vec{k}}{7}\)
\(\text{The unit vector is } \frac{6}{7}\vec{i}+\frac{2}{7}\vec{j}+\frac{3}{7}\vec{k}\)
\(\textbf{14)}\) Find the unit vector in the same direction as \(\vec{y}=-2\vec{i}+6\vec{j}-9\vec{k}\).
\(\text{The unit vector is } -\frac{2}{11}\vec{i}+\frac{6}{11}\vec{j}-\frac{9}{11}\vec{k}\)
\(|\vec{y}|=\sqrt{(-2)^2+6^2+(-9)^2}\)
\(|\vec{y}|=\sqrt{4+36+81}\)
\(|\vec{y}|=11\)
\(\text{Unit vector is }\displaystyle\frac{\vec{y}}{|\vec{y}|}\)
\(\text{Unit vector is }\displaystyle\frac{-2\vec{i}+6\vec{j}-9\vec{k}}{11}\)
\(\text{The unit vector is } -\frac{2}{11}\vec{i}+\frac{6}{11}\vec{j}-\frac{9}{11}\vec{k}\)
\(\textbf{15)}\) Find the unit vector in the same direction as \(\vec{z}=\langle 0,5,-12\rangle\).
\(\text{The unit vector is } \left\langle0,\frac{5}{13},-\frac{12}{13}\right\rangle\)
\(|\vec{z}|=\sqrt{0^2+5^2+(-12)^2}\)
\(|\vec{z}|=\sqrt{0+25+144}\)
\(|\vec{z}|=13\)
\(\text{Unit vector is }\displaystyle\frac{\vec{z}}{|\vec{z}|}\)
\(\text{Unit vector is }\displaystyle\frac{\langle0,5,-12\rangle}{13}\)
\(\text{The unit vector is } \left\langle0,\frac{5}{13},-\frac{12}{13}\right\rangle\)
Challenge Problems
\(\textbf{16)}\) Find the unit vector in the same direction as the vector from \(A(1,2)\) to \(B(7,10)\).
\(\text{The unit vector is } \left\langle\frac{3}{5},\frac{4}{5}\right\rangle\)
\(\overrightarrow{AB}=\langle7-1,10-2\rangle\)
\(\overrightarrow{AB}=\langle6,8\rangle\)
\(|\overrightarrow{AB}|=\sqrt{6^2+8^2}=10\)
\(\text{Unit vector is }\displaystyle\frac{\langle6,8\rangle}{10}\)
\(\text{The unit vector is } \left\langle\frac{3}{5},\frac{4}{5}\right\rangle\)
\(\textbf{17)}\) Find the unit vector in the same direction as the vector from \(A(-2,5)\) to \(B(10,0)\).
\(\text{The unit vector is } \left\langle\frac{12}{13},-\frac{5}{13}\right\rangle\)
\(\overrightarrow{AB}=\langle10-(-2),0-5\rangle\)
\(\overrightarrow{AB}=\langle12,-5\rangle\)
\(|\overrightarrow{AB}|=\sqrt{12^2+(-5)^2}=13\)
\(\text{Unit vector is }\displaystyle\frac{\langle12,-5\rangle}{13}\)
\(\text{The unit vector is } \left\langle\frac{12}{13},-\frac{5}{13}\right\rangle\)
\(\textbf{18)}\) Find the unit vector in the same direction as the vector from \(A(1,-1,2)\) to \(B(5,2,14)\).
\(\text{The unit vector is } \left\langle\frac{4}{13},\frac{3}{13},\frac{12}{13}\right\rangle\)
\(\overrightarrow{AB}=\langle5-1,2-(-1),14-2\rangle\)
\(\overrightarrow{AB}=\langle4,3,12\rangle\)
\(|\overrightarrow{AB}|=\sqrt{4^2+3^2+12^2}=13\)
\(\text{Unit vector is }\displaystyle\frac{\langle4,3,12\rangle}{13}\)
\(\text{The unit vector is } \left\langle\frac{4}{13},\frac{3}{13},\frac{12}{13}\right\rangle\)
\(\textbf{19)}\) Find the unit vector in the same direction as the vector from \(A(3,4,-1)\) to \(B(-3,12,23)\).
\(\text{The unit vector is } \left\langle-\frac{3}{13},\frac{4}{13},\frac{12}{13}\right\rangle\)
\(\overrightarrow{AB}=\langle-3-3,12-4,23-(-1)\rangle\)
\(\overrightarrow{AB}=\langle-6,8,24\rangle\)
\(|\overrightarrow{AB}|=\sqrt{(-6)^2+8^2+24^2}=26\)
\(\text{Unit vector is }\displaystyle\frac{\langle-6,8,24\rangle}{26}\)
\(\text{The unit vector is } \left\langle-\frac{3}{13},\frac{4}{13},\frac{12}{13}\right\rangle\)
\(\textbf{20)}\) Find a vector with magnitude \(10\) in the same direction as \(\vec{v}=\langle3,4\rangle\).
\(\text{The vector is } \langle6,8\rangle\)
\(|\vec{v}|=\sqrt{3^2+4^2}=5\)
\(\text{The unit vector in the same direction is } \left\langle\frac{3}{5},\frac{4}{5}\right\rangle\)
\(\text{Multiply the unit vector by }10.\)
\(10\left\langle\frac{3}{5},\frac{4}{5}\right\rangle=\langle6,8\rangle\)
\(\text{The vector is } \langle6,8\rangle\)
See Related Pages\(\)
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