1.4 Vektorin komponentit
https://peda.net/id/1e26b44e67e
2019-04-26T08:21:17+03:00
https://peda.net/:static/535/peda.net.logo.bg.svg
<div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div>
171
https://peda.net/id/8fdf71c867e
2019-04-26T09:53:46+03:00
a)<br/>
ei ole mahdollista, vektori voidaan jakaa erisuuntaisten vektoreiden suuntaisiin komponentteihin vain yhdellä tavalla<br/>
b)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D3%5Coverline%7Bu%7D-%5Coverline%7Bv%7D%5C%20%3Dx%5Coverline%7Bb%7D%2By%5Coverline%7Bc%7D" alt="\overline{a}=3\overline{u}-\overline{v}\ =x\overline{b}+y\overline{c}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D2%5Coverline%7Bu%7D%2B%5Coverline%7Bv%7D" alt="\overline{b}=2\overline{u}+\overline{v}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bc%7D%3D%5Coverline%7Bu%7D%2B2%5Coverline%7Bv%7D" alt="\overline{c}=\overline{u}+2\overline{v}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3%5Coverline%7Bu%7D-%5Coverline%7Bv%7D%3Dx%5Cleft(2%5Coverline%7Bu%7D%2B%5Coverline%7Bv%7D%5Cright)%2By%5Cleft(%5Coverline%7Bu%7D%2B2%5Coverline%7Bv%7D%5Cright)" alt="3\overline{u}-\overline{v}=x\left(2\overline{u}+\overline{v}\right)+y\left(\overline{u}+2\overline{v}\right)"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Coverline%7Bu%7D-%5Coverline%7Bv%7D%3D%5Cleft(2x%2By%5Cright)r%2B%5Cleft(x%2B2y%5Cright)t" alt="3\overline{u}-\overline{v}=\left(2x+y\right)r+\left(x+2y\right)t"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2b%2Bc%3D3%26%5C%5C%0Ab%2B2c%3D-1%26%5Ccdot%5Cleft(-2%5Cright)%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2b+c=3&\\ b+2c=-1&\cdot\left(-2\right) \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2b%2Bc%3D3%5C%5C%0A-2b-4c%3D2%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2b+c=3\\ -2b-4c=2 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-3c%3D5%5Cparallel%3A%5Cleft(-3%5Cright)" alt="-3c=5\parallel:\left(-3\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=c%3D-1%5C%20%5Cfrac%7B2%7D%7B3%7D" alt="c=-1\ \frac{2}{3}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2b-%5Cfrac%7B5%7D%7B3%7D%3D3%5C%5C%0Ab-%5Cfrac%7B10%7D%7B3%7D%3D-1%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2b-\frac{5}{3}=3\\ b-\frac{10}{3}=-1 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3b-5%3D2" alt="3b-5=2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3b%3D7%5C%20%5Cparallel%3A3" alt="3b=7\ \parallel:3"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=b%3D%5Cfrac%7B7%7D%7B3%7D" alt="b=\frac{7}{3}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ab%3D%5Cfrac%7B7%7D%7B3%7D%5C%5C%0Ac%3D-%5Cfrac%7B5%7D%7B3%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} b=\frac{7}{3}\\ c=-\frac{5}{3} \end{cases}"/></div>
2019-04-26T09:50:21+03:00
166
https://peda.net/id/89d48e1467e
2019-04-26T09:35:52+03:00
<img src="https://math-demo.abitti.fi/math.svg?latex=2r%5Coverline%7Ba%7D%2Br%5Coverline%7Bb%7D%2Bt%5Coverline%7Ba%7D%2Bt%5Coverline%7Bb%7D%3D5%5Coverline%7Ba%7D%2B2%5Coverline%7Bb%7D" alt="2r\overline{a}+r\overline{b}+t\overline{a}+t\overline{b}=5\overline{a}+2\overline{b}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=5%5Coverline%7Ba%7D%2B2%5Coverline%7Bb%7D%3D%5Cleft(2r%2Bt%5Cright)%5Coverline%7Ba%7D%2B%5Cleft(r%2Bt%5Cright)%5Coverline%7Bb%7D" alt="5\overline{a}+2\overline{b}=\left(2r+t\right)\overline{a}+\left(r+t\right)\overline{b}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2r%2Bt%3D5%26%5C%5C%0Ar%2Bt%3D2%26%5Ccdot%5Cleft(-1%5Cright)%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2r+t=5&\\ r+t=2&\cdot\left(-1\right) \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2r%2Bt%3D5%5C%5C%0A-r-t%3D-2%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2r+t=5\\ -r-t=-2 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=2r-r%3D5-2" alt="2r-r=5-2"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D3" alt="r=3"/> </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2%5Ccdot3%2Bt%3D5%5C%5C%0A3%2Bt%3D2%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2\cdot3+t=5\\ 3+t=2 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=2t%2B9%3D7" alt="2t+9=7"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=2t%3D-2%5C%20%5Cparallel%3A2" alt="2t=-2\ \parallel:2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=t%3D-1" alt="t=-1"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ar%3D3%5C%5C%0At%3D-1%0A%5Cend%7Bcases%7D" alt="\begin{cases} r=3\\ t=-1 \end{cases}"/></div>
2019-04-26T09:35:52+03:00
165
https://peda.net/id/7b36127a67e
2019-04-26T09:28:18+03:00
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D2%5Coverline%7Bu%7D%2B3%5Coverline%7Bv%7D" alt="\overline{a}=2\overline{u}+3\overline{v}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D%5Coverline%7Bu%7D-2%5Coverline%7Bv%7D" alt="\overline{b}=\overline{u}-2\overline{v}"/>
2019-04-26T09:28:18+03:00
168
https://peda.net/id/1aa2b9fe67e
2019-04-26T09:25:36+03:00
a)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2a%2Bb%3D1%26%5Ccdot%5Cleft(-5%5Cright)%5C%5C%0A7a%2B5b%3D14%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2a+b=1&\cdot\left(-5\right)\\ 7a+5b=14& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-10a-5b%3D-5%5C%5C%0A7a%2B5b%3D14%0A%5Cend%7Bcases%7D" alt="\begin{cases} -10a-5b=-5\\ 7a+5b=14 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-3a%3D9%5Cparallel%3A%5Cleft(-3%5Cright)" alt="-3a=9\parallel:\left(-3\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-3" alt="a=-3"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A2%5Cleft(-3%5Cright)%2Bb%3D1%5C%5C%0A7%5Cleft(-3%5Cright)%2B5b%3D14%0A%5Cend%7Bcases%7D" alt="\begin{cases} 2\left(-3\right)+b=1\\ 7\left(-3\right)+5b=14 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-6%2Bb%3D1%5C%5C%0A-21%2B5b%3D14%0A%5Cend%7Bcases%7D" alt="\begin{cases} -6+b=1\\ -21+5b=14 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ab%3D7%5C%5C%0A5b%3D35%0A%5Cend%7Bcases%7D" alt="\begin{cases} b=7\\ 5b=35 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Aa%3D-3%5C%5C%0Ab%3D7%0A%5Cend%7Bcases%7D" alt="\begin{cases} a=-3\\ b=7 \end{cases}"/><br/>
<br/>
b)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-2x%2B4y%3D-4%26%5Ccdot3%5C%5C%0A3x-6y%3D6%26%5Ccdot2%0A%5Cend%7Bcases%7D" alt="\begin{cases} -2x+4y=-4&\cdot3\\ 3x-6y=6&\cdot2 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-6x%2B12y%3D-12%5C%5C%0A6x-12y%3D12%0A%5Cend%7Bcases%7D" alt="\begin{cases} -6x+12y=-12\\ 6x-12y=12 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=12y-12y%3D12-12" alt="12y-12y=12-12"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3D0" alt="y=0"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-2x%2B4%5Ccdot0%3D-4%5C%5C%0A3x-6%5Ccdot0%3D6%0A%5Cend%7Bcases%7D" alt="\begin{cases} -2x+4\cdot0=-4\\ 3x-6\cdot0=6 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-2x%3D-4%5C%5C%0A3x%3D6%0A%5Cend%7Bcases%7D" alt="\begin{cases} -2x=-4\\ 3x=6 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3x-2x%3D6-4" alt="3x-2x=6-4"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D2%5C%5C%0Ay%3D0%5C%5C%0A%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=2\\ y=0\\ \end{cases}"/>
2019-04-26T09:25:36+03:00
163
https://peda.net/id/b76377f467e
2019-04-26T09:08:31+03:00
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D3%5Coverline%7Bv%7D%2B2%5Coverline%7Bu%7D" alt="\overline{a}=3\overline{v}+2\overline{u}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D-5%5Coverline%7Bv%7D%2B2%5Coverline%7Bu%7D" alt="\overline{b}=-5\overline{v}+2\overline{u}"/>
2019-04-26T09:08:31+03:00
169
https://peda.net/id/2d43fa9e67e
2019-04-26T09:06:43+03:00
a)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A3x%2B4y%3D4%26%5Ccdot2%5C%5C%0A-6x%2B2y%3D-3%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} 3x+4y=4&\cdot2\\ -6x+2y=-3& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A6x%2B8y%3D8%5C%5C%0A-6x%2B2y%3D-3%0A%5Cend%7Bcases%7D" alt="\begin{cases} 6x+8y=8\\ -6x+2y=-3 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=10y%3D5%5Cparallel%3A10" alt="10y=5\parallel:10"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Cfrac%7B1%7D%7B2%7D" alt="y=\frac{1}{2}"/><!--filtered attribute: style="display: inline;"--><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A3x%2B2%3D4%5C%5C%0A-6x%2B1%3D-3%0A%5Cend%7Bcases%7D" alt="\begin{cases} 3x+2=4\\ -6x+1=-3 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-3x%2B3%3D1" alt="-3x+3=1"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-3x%3D-2%5C%20%5Cparallel%3A%5Cleft(-3%5Cright)" alt="-3x=-2\ \parallel:\left(-3\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%7D%7B3%7D" alt="x=\frac{2}{3}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D%5Cfrac%7B2%7D%7B3%7D%5C%5C%0Ay%3D%5Cfrac%7B1%7D%7B2%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=\frac{2}{3}\\ y=\frac{1}{2} \end{cases}"/><br/>
<br/>
b)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A3t-r%3D%5Cfrac%7B3%7D%7B2%7D%5C%20%26%5Ccdot2%5C%5C%0A2t%2B%5Cfrac%7B2%7D%7B3%7Dr%3D1%26%5Ccdot%5Cleft(-3%5Cright)%0A%5Cend%7Bcases%7D" alt="\begin{cases} 3t-r=\frac{3}{2}\ &\cdot2\\ 2t+\frac{2}{3}r=1&\cdot\left(-3\right) \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A6t-2r%3D3%5C%5C%0A-6t-2r%3D-3%0A%5Cend%7Bcases%7D" alt="\begin{cases} 6t-2r=3\\ -6t-2r=-3 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-4r%3D0" alt="-4r=0"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=r%3D0" alt="r=0"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A3t-0%3D%5Cfrac%7B3%7D%7B2%7D%5C%5C%0A2t%2B%5Cfrac%7B2%7D%7B3%7D%5Ccdot0%3D1%0A%5Cend%7Bcases%7D" alt="\begin{cases} 3t-0=\frac{3}{2}\\ 2t+\frac{2}{3}\cdot0=1 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A3t%3D%5Cfrac%7B3%7D%7B2%7D%5C%5C%0A2t%3D1%0A%5Cend%7Bcases%7D" alt="\begin{cases} 3t=\frac{3}{2}\\ 2t=1 \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=5t%3D%5Cfrac%7B5%7D%7B2%7D%5C%20%5Cparallel%3A5" alt="5t=\frac{5}{2}\ \parallel:5"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=t%3D%5Cfrac%7B1%7D%7B2%7D" alt="t=\frac{1}{2}"/><br/>
<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0At%3D%5Cfrac%7B1%7D%7B2%7D%5C%5C%0Ar%3D0%0A%5Cend%7Bcases%7D" alt="\begin{cases} t=\frac{1}{2}\\ r=0 \end{cases}"/></div>
2019-04-26T09:04:39+03:00