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What are the characteristics of the letters that are commonly used to represent vectors?
lowercase, bold letter, usually $\text{\hspace{0.17em}}u,v,w$
How is a vector more specific than a line segment?
What are $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j,$ and what do they represent?
They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.
What is component form?
When a unit vector is expressed as $\u27e8a,b\u27e9,$ which letter is the coefficient of the $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and which the $\text{\hspace{0.17em}}j?$
The first number always represents the coefficient of the $\text{\hspace{0.17em}}i,\text{\hspace{0.17em}}$ and the second represents the $\text{\hspace{0.17em}}j.$
Given a vector with initial point $\text{\hspace{0.17em}}\left(5,2\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}\left(-1,-3\right),\text{\hspace{0.17em}}$ find an equivalent vector whose initial point is $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ Write the vector in component form $\u27e8a,b\u27e9.$
Given a vector with initial point $\text{\hspace{0.17em}}\left(-4,2\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}\left(3,-3\right),\text{\hspace{0.17em}}$ find an equivalent vector whose initial point is $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ Write the vector in component form $\u27e8a,b\u27e9.$
$\u30087,-5\u3009$
Given a vector with initial point $\text{\hspace{0.17em}}\left(7,-1\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}\left(-1,-7\right),\text{\hspace{0.17em}}$ find an equivalent vector whose initial point is $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ Write the vector in component form $\u27e8a,b\u27e9.$
For the following exercises, determine whether the two vectors $\text{\hspace{0.17em}}u\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ are equal, where $\text{\hspace{0.17em}}u\text{\hspace{0.17em}}$ has an initial point $\text{\hspace{0.17em}}{P}_{1}\text{\hspace{0.17em}}$ and a terminal point $\text{\hspace{0.17em}}{P}_{2}\text{\hspace{0.17em}}$ and $v$ has an initial point $\text{\hspace{0.17em}}{P}_{3}\text{\hspace{0.17em}}$ and a terminal point $\text{\hspace{0.17em}}{P}_{4}$ .
${P}_{1}=\left(5,1\right),{P}_{2}=\left(3,-2\right),{P}_{3}=\left(-1,3\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{P}_{4}=\left(9,-4\right)$
not equal
${P}_{1}=\left(2,-3\right),{P}_{2}=\left(5,1\right),{P}_{3}=\left(6,-1\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{P}_{4}=\left(9,3\right)$
${P}_{1}=\left(-1,-1\right),{P}_{2}=\left(-4,5\right),{P}_{3}=\left(-10,6\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{P}_{4}=\left(-13,12\right)$
equal
${P}_{1}=\left(3,7\right),{P}_{2}=\left(2,1\right),{P}_{3}=\left(1,2\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{P}_{4}=\left(-1,-4\right)$
${P}_{1}=\left(8,3\right),{P}_{2}=\left(6,5\right),{P}_{3}=\left(11,8\right),\text{\hspace{0.17em}}$ and ${P}_{4}=\left(9,10\right)$
equal
Given initial point $\text{\hspace{0.17em}}{P}_{1}=\left(-3,1\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}{P}_{2}=\left(5,2\right),\text{\hspace{0.17em}}$ write the vector $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j.\text{\hspace{0.17em}}$
Given initial point $\text{\hspace{0.17em}}{P}_{1}=\left(6,0\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}{P}_{2}=\left(-1,-3\right),\text{\hspace{0.17em}}$ write the vector $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j.\text{\hspace{0.17em}}$
$7i-3j$
For the following exercises, use the vectors u = i + 5 j , v = −2 i − 3 j , and w = 4 i − j .
Find u + ( v − w )
For the following exercises, use the given vectors to compute u + v , u − v , and 2 u − 3 v .
$u=\u27e82,-3\u27e9,v=\u27e81,5\u27e9$
$u=\u27e8-3,4\u27e9,v=\u27e8-2,1\u27e9$
$u+v=\u3008-5,5\u3009,u-v=\u3008-1,3\u3009,2u-3v=\u30080,5\u3009$
Let v = −4 i + 3 j . Find a vector that is half the length and points in the same direction as $\text{\hspace{0.17em}}v.$
Let v = 5 i + 2 j . Find a vector that is twice the length and points in the opposite direction as $\text{\hspace{0.17em}}v.$
$-10i\u20134j$
For the following exercises, find a unit vector in the same direction as the given vector.
a = 3 i + 4 j
b = −2 i + 5 j
$-\frac{2\sqrt{29}}{29}i+\frac{5\sqrt{29}}{29}j$
c = 10 i – j
$d=-\frac{1}{3}i+\frac{5}{2}j$
$-\frac{2\sqrt{229}}{229}i+\frac{15\sqrt{229}}{229}j$
u = 100 i + 200 j
u = −14 i + 2 j
$-\frac{7\sqrt{2}}{10}i+\frac{\sqrt{2}}{10}j$
For the following exercises, find the magnitude and direction of the vector, $\text{\hspace{0.17em}}0\le \theta <2\pi .$
$\u27e80,4\u27e9$
$\u27e82,\mathrm{-5}\u27e9$
$\u27e8\mathrm{-4},\mathrm{-6}\u27e9$
$\left|v\right|=7.211,\theta =\mathrm{236.310\xb0}$
Given u = 3 i − 4 j and v = −2 i + 3 j , calculate $\text{\hspace{0.17em}}u\cdot v.$
Given u = − i − j and v = i + 5 j , calculate $\text{\hspace{0.17em}}u\cdot v.$
$-6$
Given $\text{\hspace{0.17em}}u=\u27e8-2,4\u27e9\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}v=\u27e8-3,1\u27e9,\text{\hspace{0.17em}}$ calculate $\text{\hspace{0.17em}}u\cdot v.$
Given u $=\u27e8-1,6\u27e9$ and v $=\u27e86,-1\u27e9,$ calculate $\text{\hspace{0.17em}}u\cdot v.$
$-12$
For the following exercises, given $\text{\hspace{0.17em}}v,\text{\hspace{0.17em}}$ draw $v,$ 3 v and $\text{\hspace{0.17em}}\frac{1}{2}v.$
$\u27e82,\mathrm{-1}\u27e9$
$\u27e8\mathrm{-3},\mathrm{-2}\u27e9$
For the following exercises, use the vectors shown to sketch u + v , u − v , and 2 u .
For the following exercises, use the vectors shown to sketch 2 u + v .
For the following exercises, use the vectors shown to sketch u − 3 v .
For the following exercises, write the vector shown in component form.
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