4.1 Suora tasossa ja avaruudessa
https://peda.net/id/267ec7da779
2019-05-16T08:33:35+03:00
https://peda.net/:static/537/peda.net.logo.bg.svg
<div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div>
412
https://peda.net/id/875f3dfa786
2019-05-17T09:39:32+03:00
a)
2019-05-17T09:39:32+03:00
413
https://peda.net/id/e688ac36786
2019-05-17T09:35:16+03:00
suora leikkaa xy-tason pisteessä (-6,9,0)<br/>
suora leikkaa z-akselin pisteessä (0,0,6)<a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4stja/413/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4stja/413/sieppaa-png:file/photo/6bca1c5eafe49ed6ab6ae49d1e4f2cd8e5bb8591/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a>
2019-05-17T09:35:02+03:00
421
https://peda.net/id/33055c22786
2019-05-17T09:30:01+03:00
<div>421</div>
<div>a)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D%5Cleft(-4%5Cright)%5E2%2B3%5E2" alt="x^2=\left(-4\right)^2+3^2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D5" alt="x=5"/></div>
<div>b)</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D12%5E2%2B5%5E2" alt="x^2=12^2+5^2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D13" alt="x=13"/></div>
2019-05-17T09:30:01+03:00
422
https://peda.net/id/8d2d906c786
2019-05-17T09:25:23+03:00
<img src="https://math-demo.abitti.fi/math.svg?latex=1%5C%20asteikon%5C%20ruutu%5C%20kuvaa%5C%201km%5E2" alt="1\ asteikon\ ruutu\ kuvaa\ 1km^2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=100km-50km%3D50km" alt="100km-50km=50km"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B50%7D%7B5%7D%3D10" alt="\frac{50}{5}=10"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=meteoriitti%5C%20kulkee%5C%2010%5C%20koordinaatiston%5C%20v%C3%A4li%C3%A4" alt="meteoriitti\ kulkee\ 10\ koordinaatiston\ väliä"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=10%5Cleft(%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D%5Cright)%3D10%5Coverline%7B%5Ctext%7Bi%7D%7D%2B30%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="10\left(\overline{\text{i}}+3\overline{\text{j}}\right)=10\overline{\text{i}}+30\overline{\text{j}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C10%5Coverline%7B%5Ctext%7Bi%7D%7D%2B30%5Coverline%7B%5Ctext%7Bj%7D%7D%5Cright%7C%3D10%5Csqrt%7B10%7D%5Capprox31%7B%2C%7D622...km" alt="\left|10\overline{\text{i}}+30\overline{\text{j}}\right|=10\sqrt{10}\approx31{,}622...km"/>
2019-05-17T09:25:23+03:00
420
https://peda.net/id/c66fc2d8786
2019-05-17T09:19:49+03:00
<div>a) kohtisuorassa xy-tasoa vastaan<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D1%26%5C%5C%0Ay%3D2%26%5C%5C%0Az%3D3t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=1&\\ y=2&\\ z=3t& \end{cases}"/></div>
<div>b) yhdensuuntainen y-akselin kanssa<br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D1%26%5C%5C%0Ay%3D2t%26%5C%5C%0Az%3D3%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=1&\\ y=2t&\\ z=3& \end{cases}"/></div>
c) yhdensuuntainen pisteiden (1,0,1) ja (2,1,3) kautta kulkevan suoran kanssa<br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D1%2Bt%26%5C%5C%0Ay%3D2%2Bt%26%5C%5C%0Az%3D3%2B2t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=1+t&\\ y=2+t&\\ z=3+2t& \end{cases}"/></div>
<div> </div>
</div>
2019-05-17T09:19:49+03:00
419
https://peda.net/id/0cc526ee786
2019-05-17T09:08:14+03:00
<a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4stja/419/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4stja/419/sieppaa-png:file/photo/9e5df0b668481b71a4e37323099b962816ae7b3f/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/>
<br/>
kaikki yhtälöt A-D esittävät suoraa a, koska ne alkavat samasta pisteestä ja niiden muodostamien suorien suuntavektorit ovat yhdensuuntaiset
2019-05-17T09:07:28+03:00
414
https://peda.net/id/445ff7d8786
2019-05-17T09:01:52+03:00
a)<br/>
suorat voivat myös olla yhdensuuntaiset tai ristikkäiset, jolloin ne eivät leikkaa<br/>
esimerkiksi nämä avaruuden suorat ovat yhdensuuntaiset, eivätkä leikkaa<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D0%26%5C%5C%0Ay%3D0%26%5C%5C%0Az%3Dt%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=0&\\ y=0&\\ z=t& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D1%26%5C%5C%0Ay%3D0%26%5C%5C%0Az%3Dt%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=1&\\ y=0&\\ z=t& \end{cases}"/><br/>
b)<br/>
suora voi myös kulkea xy-tason suuntaisesti koskematta tasoon, esimerkiksi<br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D1%26%5C%5C%0Ay%3Dt%26%5C%5C%0Az%3D1%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=1&\\ y=t&\\ z=1& \end{cases}"/><br/>
c)<br/>
suoran ei ole pakko leikata mitään akselia, vaan se voi olla niiden kanssa ristikkäinen, esimerkiksi<br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D2t%2B1%26%5C%5C%0Ay%3Dt%26%5C%5C%0Az%3Dt%2B1%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=2t+1&\\ y=t&\\ z=t+1& \end{cases}"/></div>
</div>
2019-05-17T09:01:52+03:00
411
https://peda.net/id/80c4a6aa77a
2019-05-17T08:50:14+03:00
a)<br/>
<a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4stja/411/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4stja/411/sieppaa-png:file/photo/9985af75fa50779bc292642697aeb80df007b8fb/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/>
b)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bs%7D%3D3%5Coverline%7B%5Ctext%7Bi%7D%7D-2%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{s}=3\overline{\text{i}}-2\overline{\text{j}}"/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-5%2B3t%26%5C%5C%0Ay%3D4-2t%26t%5Cin%5Cmathbb%7BR%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=-5+3t&\\ y=4-2t&t\in\mathbb{R} \end{cases}"/></div>
<img src="https://math-demo.abitti.fi/math.svg?latex=b%3D2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="b=2\overline{\text{i}}+\overline{\text{j}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D6%2B2r%26%5C%5C%0Ay%3D3%2Br%26r%5Cin%5Cmathbb%7BR%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=6+2r&\\ y=3+r&r\in\mathbb{R} \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-5%2B3t%3D6%2B2r%26%5C%5C%0A4-2t%3D3%2Br%26%5Cparallel%5Ccdot%5Cleft(-2%5Cright)%0A%5Cend%7Bcases%7D" alt="\begin{cases} -5+3t=6+2r&\\ 4-2t=3+r&\parallel\cdot\left(-2\right) \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-5%2B3t%3D6%2B2r%26%5C%5C%0A-8%2B4t%3D-6-2r%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} -5+3t=6+2r&\\ -8+4t=-6-2r& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=7t%3D13%7B%2C%7D%5C%20t%3D%5Cfrac%7B13%7D%7B7%7D" alt="7t=13{,}\ t=\frac{13}{7}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=ratkaistaan%5C%20r%5C%20sijoittamalla%5C%20t%5C%20yht%C3%A4l%C3%B6pariin" alt="ratkaistaan\ r\ sijoittamalla\ t\ yhtälöpariin"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-5%2B3%5Ccdot%5Cfrac%7B13%7D%7B7%7D%3D6%2B2r%26%5C%5C%0A4-2%5Ccdot%5Cfrac%7B13%7D%7B7%7D%3D3%2Br%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} -5+3\cdot\frac{13}{7}=6+2r&\\ 4-2\cdot\frac{13}{7}=3+r& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-%5Cfrac%7B38%7D%7B7%7D%3D2r%26%5C%5C%0A-%5Cfrac%7B19%7D%7B7%7D%3Dr%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} -\frac{38}{7}=2r&\\ -\frac{19}{7}=r& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=3r%3D-%5Cfrac%7B57%7D%7B7%7D%7B%2C%7D%5C%20r%3D-%5Cfrac%7B19%7D%7B7%7D" alt="3r=-\frac{57}{7}{,}\ r=-\frac{19}{7}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=sijoitetaan%5C%20t%5C%20suoran%5C%20a%5C%20yht%C3%A4l%C3%B6%C3%B6n" alt="sijoitetaan\ t\ suoran\ a\ yhtälöön"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-5%2B%5Cfrac%7B39%7D%7B7%7D%26%5C%5C%0Ay%3D4-%5Cfrac%7B26%7D%7B7%7D%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=-5+\frac{39}{7}&\\ y=4-\frac{26}{7}& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D%5Cfrac%7B4%7D%7B7%7D%26%5C%5C%0Ay%3D%5Cfrac%7B2%7D%7B7%7D%26%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=\frac{4}{7}&\\ y=\frac{2}{7}& \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=suorat%5C%20a%5C%20ja%5C%20b%5C%20leikkaavat%5C%20pisteess%C3%A4%5C%20%5Cleft(%5Cfrac%7B4%7D%7B7%7D%7B%2C%7D%5C%20%5Cfrac%7B2%7D%7B7%7D%5Cright)" alt="suorat\ a\ ja\ b\ leikkaavat\ pisteessä\ \left(\frac{4}{7}{,}\ \frac{2}{7}\right)"/><br/>
<br/>
<br/>
2019-05-16T09:54:51+03:00
410
https://peda.net/id/d7b5306677a
2019-05-16T09:50:07+03:00
a)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(-53%7B%2C%7D%5C%2018%7B%2C%7D%5C%2071%5Cright)" alt="A=\left(-53{,}\ 18{,}\ 71\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=OA%3DOP%2Bt%5Coverline%7Bv%7D" alt="OA=OP+t\overline{v}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=P%3D%5Cleft(10%7B%2C%7D-27%7B%2C%7D7%5Cright)" alt="P=\left(10{,}-27{,}7\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bv%7D%3D7%5Coverline%7B%5Ctext%7Bi%7D%7D-5%5Coverline%7B%5Ctext%7Bj%7D%7D-8%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{v}=7\overline{\text{i}}-5\overline{\text{j}}-8\overline{\text{k}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-53%5Coverline%7B%5Ctext%7Bi%7D%7D%2B18%5Coverline%7B%5Ctext%7Bj%7D%7D%2B71%5Coverline%7B%5Ctext%7Bk%7D%7D%3D10%5Coverline%7B%5Ctext%7Bi%7D%7D-27%5Coverline%7B%5Ctext%7Bj%7D%7D%2B7%5Coverline%7B%5Ctext%7Bk%7D%7D%2B7t%5Coverline%7B%5Ctext%7Bi%7D%7D-5t%5Coverline%7B%5Ctext%7Bj%7D%7D-8t%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="-53\overline{\text{i}}+18\overline{\text{j}}+71\overline{\text{k}}=10\overline{\text{i}}-27\overline{\text{j}}+7\overline{\text{k}}+7t\overline{\text{i}}-5t\overline{\text{j}}-8t\overline{\text{k}}"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-53%3D7t%2B10%26t%3D-9%5C%5C%0A18%3D-5t-27%26t%3D-9%5C%5C%0A71%3D-8t%2B7%26t%3D-8%0A%5Cend%7Bcases%7D" alt="\begin{cases} -53=7t+10&t=-9\\ 18=-5t-27&t=-9\\ 71=-8t+7&t=-8 \end{cases}"/></div>
<div>piste A ei ole suoralla</div>
<br/>
b)<br/>
<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=B%3D%5Cleft(2%7B%2C%7D%5C%20-4%7B%2C%7D%5C%205%5Cright)" alt="B=\left(2{,}\ -4{,}\ 5\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=C%3D%5Cleft(-8%7B%2C%7D%5C%200%7B%2C%7D%5C%2017%5Cright)" alt="C=\left(-8{,}\ 0{,}\ 17\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=suoran%5C%20suuntavektori%3D%5Coverline%7BBC%7D" alt="suoran\ suuntavektori=\overline{BC}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBC%7D%3D-10%5Coverline%7B%5Ctext%7Bi%7D%7D-4%5Coverline%7B%5Ctext%7Bj%7D%7D%2B12%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{BC}=-10\overline{\text{i}}-4\overline{\text{j}}+12\overline{\text{k}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOA%7D%3D%5Coverline%7BOB%7D%2Bt%5Coverline%7Bv%7D" alt="\overline{OA}=\overline{OB}+t\overline{v}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=-53%5Coverline%7B%5Ctext%7Bi%7D%7D%2B18%5Coverline%7B%5Ctext%7Bj%7D%7D%2B71%5Coverline%7B%5Ctext%7Bk%7D%7D%3D2%5Coverline%7B%5Ctext%7Bi%7D%7D-4%5Coverline%7B%5Ctext%7Bj%7D%7D%2B5%5Coverline%7B%5Ctext%7Bk%7D%7D-10t%5Coverline%7B%5Ctext%7Bi%7D%7D-4t%5Coverline%7B%5Ctext%7Bj%7D%7D%2B12t%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="-53\overline{\text{i}}+18\overline{\text{j}}+71\overline{\text{k}}=2\overline{\text{i}}-4\overline{\text{j}}+5\overline{\text{k}}-10t\overline{\text{i}}-4t\overline{\text{j}}+12t\overline{\text{k}}"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-53%3D2-10t%26t%3D5%7B%2C%7D5%5C%5C%0A18%3D-4-4t%26t%3D5%7B%2C%7D5%5C%5C%0A71%3D5%2B12t%26t%3D5%7B%2C%7D5%0A%5Cend%7Bcases%7D" alt="\begin{cases} -53=2-10t&t=5{,}5\\ 18=-4-4t&t=5{,}5\\ 71=5+12t&t=5{,}5 \end{cases}"/></div>
<div>Piste A on suoralla</div>
2019-05-16T09:50:07+03:00
409
https://peda.net/id/3ee76ea477a
2019-05-16T09:38:42+03:00
Mitkä vaihtoehdoista A-D esittävät suoraa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-1%2B3t%26%5C%5C%0Ay%3D2-t%26t%5Cin%5Cmathbb%7BR%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=-1+3t&\\ y=2-t&t\in\mathbb{R} \end{cases}"/><br/>
ABC
2019-05-16T09:38:41+03:00
408
https://peda.net/id/7aab701c77a
2019-05-16T09:33:12+03:00
a)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOP%7D%3D%5Coverline%7BOA%7D%2Bt%5Coverline%7Bv%7D" alt="\overline{OP}=\overline{OA}+t\overline{v}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOP%7D%3D2i-k%2Bt%5Cleft(-%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D-4%5Coverline%7B%5Ctext%7Bk%7D%7D%5Cright)" alt="\overline{OP}=2i-k+t\left(-\overline{\text{i}}+3\overline{\text{j}}-4\overline{\text{k}}\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=suuntavektori%5C%20on%5C%20-%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D-4%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="suuntavektori\ on\ -\overline{\text{i}}+3\overline{\text{j}}-4\overline{\text{k}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=jokin%5C%20suoran%5C%20piste%5C%20on%5C%20%5Cleft(2%7B%2C%7D0%7B%2C%7D-1%5Cright)" alt="jokin\ suoran\ piste\ on\ \left(2{,}0{,}-1\right)"/><br/>
b)<br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D2-t%26%5C%5C%0Ay%3D3%26%5C%5C%0Az%3D5t%26t%5Cin%5Cmathbb%7BR%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=2-t&\\ y=3&\\ z=5t&t\in\mathbb{R} \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=suuntavektori%5C%20on%5C%20%5Cleft(2-t%5Cright)%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D%2B5t%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="suuntavektori\ on\ \left(2-t\right)\overline{\text{i}}+3\overline{\text{j}}+5t\overline{\text{k}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=jokin%5C%20suoran%5C%20piste" alt="jokin\ suoran\ piste"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=suoran%5C%20piste%5C%20kun%5C%20t%3D0" alt="suoran\ piste\ kun\ t=0"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2%7B%2C%7D%5C%203%7B%2C%7D%5C%200%5Cright)" alt="\left(2{,}\ 3{,}\ 0\right)"/>
2019-05-16T09:33:12+03:00
417
https://peda.net/id/803f175077a
2019-05-16T09:26:12+03:00
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D3-2t%26%5C%5C%0Ay%3D-1%2B2t%26%5C%5C%0Az%3D3-t%26t%5Cin%5Cmathbb%7BR%7D%0A%5Cend%7Bcases%7D" alt="\begin{cases} x=3-2t&\\ y=-1+2t&\\ z=3-t&t\in\mathbb{R} \end{cases}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=suora%5C%20kulkee%5C%20pisteen%5C%20B%5Cleft(3%7B%2C%7D%5C%20-1%7B%2C%7D%5C%203%5Cright)%5C%20kautta" alt="suora\ kulkee\ pisteen\ B\left(3{,}\ -1{,}\ 3\right)\ kautta"/><br/>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=ja%5C%20suoran%5C%20suuntavektori%5C%20on%5C%20-2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B2%5Coverline%7B%5Ctext%7Bj%7D%7D-%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="ja\ suoran\ suuntavektori\ on\ -2\overline{\text{i}}+2\overline{\text{j}}-\overline{\text{k}}"/></div>
<div>Pisteen A etäisyys suorasta on kohtisuorasti mitattu etäisyys</div>
<div>toisaalta etäisyys on vektorin<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAA'%7D" alt="\overline{AA'}"/>pituus</div>
<div>Piste A' on suoralla, joten sen koordinaatit ovat muotoa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3-2t%7B%2C%7D%5C%20-1%2B2t%7B%2C%7D%5C%203-t%5Cright)%7B%2C%7D%5C%20t%5Cin%5Cmathbb%7BR%7D" alt="\left(3-2t{,}\ -1+2t{,}\ 3-t\right){,}\ t\in\mathbb{R}"/></div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAA'%7D%3D-2t%5Coverline%7B%5Ctext%7Bi%7D%7D-12%2B2t%5Coverline%7B%5Ctext%7Bj%7D%7D-6-t%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{AA'}=-2t\overline{\text{i}}-12+2t\overline{\text{j}}-6-t\overline{\text{k}}"/></div>
<div>Nyt</div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAA'%7D%5Cperp%5Coverline%7Bs%7D" alt="\overline{AA'}\perp\overline{s}"/></div>
<div>joten </div>
<div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAA'%7D%5Ccdot%5Coverline%7Bs%7D%3D0" alt="\overline{AA'}\cdot\overline{s}=0"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAA'%7D%5Ccdot%5Coverline%7Bs%7D%3D-2t%5Ccdot-2%2B%5Cleft(-12%2B2t%5Cright)%5Ccdot2%2B%5Cleft(-6-t%5Cright)%5Ccdot%5Cleft(-1%5Cright)%3D0" alt="\overline{AA'}\cdot\overline{s}=-2t\cdot-2+\left(-12+2t\right)\cdot2+\left(-6-t\right)\cdot\left(-1\right)=0"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=9t-18%3D0" alt="9t-18=0"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=t%3D2" alt="t=2"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAA'%7D%3D-4%5Coverline%7B%5Ctext%7Bi%7D%7D-8%5Coverline%7B%5Ctext%7Bj%7D%7D-8%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{AA'}=-4\overline{\text{i}}-8\overline{\text{j}}-8\overline{\text{k}}"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAA'%7D%5Cright%7C%3D%5Csqrt%7B4%5E2%2B8%5E2%2B8%5E2%7D%3D12%5C%20%5Cleft(tai%5C%20-12%5Cright)" alt="\left|\overline{AA'}\right|=\sqrt{4^2+8^2+8^2}=12\ \left(tai\ -12\right)"/><br/>
<img src="https://math-demo.abitti.fi/math.svg?latex=pisteen%5C%20A%5C%20%5Cleft(3%7B%2C%7D%5C%2011%7B%2C%7D%5C%203%5Cright)%5C%20et%C3%A4isyys%5C%20suorasta%5C%20on%5C%2012%5C%20" alt="pisteen\ A\ \left(3{,}\ 11{,}\ 3\right)\ etäisyys\ suorasta\ on\ 12\ "/></div>
2019-05-16T09:26:12+03:00