https://peda.net/id/0c3b77ed71e 2025-08-05T13:50:46+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> https://peda.net/id/0c3bde5a71e 2018-05-18T14:31:49+03:00 <span class="editor underline"><b>439.</b></span><br/> Pisteen <img src="http://laksyvihko.fi/math.svg?latex=P%3D%5Cleft(6%7B%2C%7D%5C%20-20%7B%2C%7D%5C%20%5C%2041%5Cright)" alt="P=\left(6{,}\ -20{,}\ \ 41\right)"/> etäisyys tasosta <img src="http://laksyvihko.fi/math.svg?latex=x-4y%2B8z-9%3D0" alt="x-4y+8z-9=0"/>.<br/> <br/> <div>(Tarkistetaan, onko piste P tasolla: <img src="http://laksyvihko.fi/math.svg?latex=6-4%5Ccdot%5Cleft(-20%5Cright)%2B8%5Ccdot41-9%3D405" alt="6-4\cdot\left(-20\right)+8\cdot41-9=405"/>, joten piste P ei ole tasolla)</div> <div> </div> <div> <div>Tarvitaan tasolle kohtisuora suora, joka kulkee pisteen P kautta.<br/> <br/> Suoran suuntavektori on <img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7Bn%7D%3D1%5Coverline%7Bi%7D-4%5Coverline%7Bj%7D%2B8%5Coverline%7Bk%7D" alt="\overline{n}=1\overline{i}-4\overline{j}+8\overline{k}"/>, joten parametrimuotoinen yhtälö on<br/> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D6%2Bt%26%5C%5C%0Ay%3D-20-4t%26%5C%20%5C%20%5C%20t%5Cin%5Cmathbb%7BR%7D%5C%5C%0Az%3D41%2B8t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=6+t&amp;\\&#10;y=-20-4t&amp;\ \ \ t\in\mathbb{R}\\&#10;z=41+8t&amp;&#10;\end{cases}"/><br/> <br/> </div> </div> <div>Tasolla kohtisuoraan oleva piste on <img src="http://laksyvihko.fi/math.svg?latex=P%27%3D%5Cleft(6%2Bt%7B%2C%7D%5C%20-20-4t%7B%2C%7D%5C%2041%2B8t%5Cright)" alt="P'=\left(6+t{,}\ -20-4t{,}\ 41+8t\right)"/><br/> <br/> </div> </div> <div>Sijoitetaan koordinaatit tason yhtälöön ja ratkaistaan<span> </span><em>t</em></div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Cleft(6%2Bt%5Cright)-4%5Cleft(-20-4t%5Cright)%2B8%5Cleft(41%2B8t%5Cright)-9%3D0" alt="\left(6+t\right)-4\left(-20-4t\right)+8\left(41+8t\right)-9=0"/></div> <div><img src="http://laksyvihko.fi/math.svg?latex=t%3D-5" alt="t=-5"/><br/> <br/> </div> Sijoitetaan tämä suoran yhtälöön, niin saadaan piste P'<br/> <img src="http://laksyvihko.fi/math.svg?latex=P%27%5C%20%3D%5Cleft(6%2B%5Cleft(-5%5Cright)%7B%2C%7D%5C%20-20-4%5Ccdot%5Cleft(-5%5Cright)%7B%2C%7D%5C%2041%2B8%5Ccdot%5Cleft(-5%5Cright)%5Cright)" alt="P'\ =\left(6+\left(-5\right){,}\ -20-4\cdot\left(-5\right){,}\ 41+8\cdot\left(-5\right)\right)"/><br/> <img src="http://laksyvihko.fi/math.svg?latex=%3D%5Cleft(1%7B%2C%7D%5C%200%7B%2C%7D%5C%201%5Cright)" alt="=\left(1{,}\ 0{,}\ 1\right)"/> <div> </div> <div>Lasketaan vektorin PP' pituus</div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7BPP%27%7D%3D%5Cleft(1-6%5Cright)%5C%20%5Coverline%7Bi%7D%2B%5C%20%5Cleft(0-%5Cleft(-20%5Cright)%5Cright)%5C%20%5Coverline%7Bj%7D%2B%5Cleft(1-41%5Cright)%5C%20%5Coverline%7Bk%7D" alt="\overline{PP'}=\left(1-6\right)\ \overline{i}+\ \left(0-\left(-20\right)\right)\ \overline{j}+\left(1-41\right)\ \overline{k}"/><img src="http://laksyvihko.fi/math.svg?latex=%3D-5%5Coverline%7Bi%7D%5C%20%2B%5C%2020%5C%20%5Coverline%7Bj%7D%5C%20-40%5C%20%5Coverline%7Bk%7D" alt="=-5\overline{i}\ +\ 20\ \overline{j}\ -40\ \overline{k}"/></div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BPP%27%7D%5Cright%7C%3D%5Csqrt%7B5%5E2%2B20%5E2%2B40%5E2%7D%3D45" alt="\left|\overline{PP'}\right|=\sqrt{5^2+20^2+40^2}=45"/><br/> <div><br/>  <br/> <span class="editor underline"><b>ESIMERKKI</b></span><br/> <div>Suora</div> <img src="http://laksyvihko.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D3-4t%26%5C%5C%0Ay%3D2%2Bt%26%5C%20%5C%20t%5Cin%5Cmathbb%7BR%7D%5C%5C%0Az%3D-1%2B3t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=3-4t&amp;\\&#10;y=2+t&amp;\ \ t\in\mathbb{R}\\&#10;z=-1+3t&amp;&#10;\end{cases}"/> <div>on kohtisuorassa tasoa vastaan.</div> <div> </div> <div>Määritä sen tason yhtälö, jonka eräs piste on <img src="http://laksyvihko.fi/math.svg?latex=A%3D%5Cleft(2%7B%2C%7D%5C%204%7B%2C%7D%5C%203%5Cright)" alt="A=\left(2{,}\ 4{,}\ 3\right)"/>.</div> <div>--------------------</div> <div>Tason (eräs) normaalivektori on annetun suoran suuntavektori (koska suora on kohtisuorassa tasoa vastaan):</div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7Bn%7D%3D-4%5Coverline%7Bi%7D%2B1%5Coverline%7Bj%7D%2B3%5Coverline%7Bk%7D" alt="\overline{n}=-4\overline{i}+1\overline{j}+3\overline{k}"/></div> <span>joten tason yhtälö on:</span><br/> <img src="http://laksyvihko.fi/math.svg?latex=a%5Ccdot%5Cleft(x-x_0%5Cright)%2Bb%5Ccdot%5Cleft(y-y_0%5Cright)%2Bc%5Ccdot%5Cleft(z-z_0%5Cright)%3D0" alt="a\cdot\left(x-x_0\right)+b\cdot\left(y-y_0\right)+c\cdot\left(z-z_0\right)=0"/><br/> <div><img src="http://laksyvihko.fi/math.svg?latex=-4%5Ccdot%5Cleft(x-2%5Cright)%2B1%5Ccdot%5Cleft(y-4%5Cright)%2B3%5Ccdot%5Cleft(z-3%5Cright)%3D0" alt="-4\cdot\left(x-2\right)+1\cdot\left(y-4\right)+3\cdot\left(z-3\right)=0"/></div> <img src="http://laksyvihko.fi/math.svg?latex=-4x%2B8%2By-4%2B3z-9%3D0" alt="-4x+8+y-4+3z-9=0"/><br/> <img src="http://laksyvihko.fi/math.svg?latex=-4x%2By%2B3z-5%3D0" alt="-4x+y+3z-5=0"/><br/> <div> </div> <div><span class="editor underline">V: Tason yhtälö on .</span></div> <br/> <b>440.</b><br/> <div>a) (1, <b>0</b>, <b>0</b>)</div> <div>b) (1, 2, <b>0</b>)</div> <div> </div> <span>c)</span> <div>P = (1, 2, 3)<br/> <div>Muodostetaan tasolle kohtisuoran suoran, joka kulkee pisteen P kautta, parametrimuotoinen yhtälö</div> <div>suoran suuntavektori on tason normaalivektori <img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7Bn%7D%3D2%5Coverline%7Bi%7D%2B3%5Coverline%7Bj%7D%5C%20-%5Coverline%7Bk%7D" alt="\overline{n}=2\overline{i}+3\overline{j}\ -\overline{k}"/></div> </div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D1%2B2t%26%5C%5C%0Ay%3D2%2B3t%26%5C%20%5C%20%5C%20t%5Cin%5Cmathbb%7BR%7D%5C%5C%0Az%3D3-t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=1+2t&amp;\\&#10;y=2+3t&amp;\ \ \ t\in\mathbb{R}\\&#10;z=3-t&amp;&#10;\end{cases}"/></div> <div>Projektiopisteen P' koordinaatit ovat x = 1+2t, y = 2+3t, z= 3-t.</div> <div>Sijoitetaan nämä tason yhtälöön:</div> <div><img src="http://laksyvihko.fi/math.svg?latex=2%5Cleft%281%2B2t%5Cright%29%2B3%5Cleft%282%2B3t%5Cright%29-%5Cleft%283-t%5Cright%29%2B1%3D0" alt="2\left(1+2t\right)+3\left(2+3t\right)-\left(3-t\right)+1=0"/></div> <img src="http://laksyvihko.fi/math.svg?latex=2%2B4t%2B6%2B9t-3%2Bt%2B1%3D0" alt="2+4t+6+9t-3+t+1=0"/><br/> <img src="http://laksyvihko.fi/math.svg?latex=14t%2B6%3D0" alt="14t+6=0"/><br/> <img src="http://laksyvihko.fi/math.svg?latex=t%3D-%5Cfrac%7B6%7D%7B14%7D%3D-%5Cfrac%7B3%7D%7B7%7D" alt="t=-\frac{6}{14}=-\frac{3}{7}"/> <div>sijoitetaan: <img src="http://laksyvihko.fi/math.svg?latex=P%27%3D%5Cleft%281%2B2%5Ccdot%5Cfrac%7B-3%7D%7B7%7D%7B%2C%7D%5C%202%2B3%5Ccdot%5Cfrac%7B-3%7D%7B7%7D%7B%2C%7D%5C%203-%5Cfrac%7B-3%7D%7B7%7D%5Cright%29" alt="P'=\left(1+2\cdot\frac{-3}{7}{,}\ 2+3\cdot\frac{-3}{7}{,}\ 3-\frac{-3}{7}\right)"/></div> <img src="http://laksyvihko.fi/math.svg?latex=%3D%5Cleft%28%5Cfrac%7B7%7D%7B7%7D%2B%5Cfrac%7B-6%7D%7B7%7D%7B%2C%7D%5C%20%5Cfrac%7B14%7D%7B7%7D%2B%5Cfrac%7B-9%7D%7B7%7D%7B%2C%7D%5C%20%5Cfrac%7B21%7D%7B7%7D-%5Cfrac%7B-3%7D%7B7%7D%5Cright%29" alt="=\left(\frac{7}{7}+\frac{-6}{7}{,}\ \frac{14}{7}+\frac{-9}{7}{,}\ \frac{21}{7}-\frac{-3}{7}\right)"/><br/> <img src="http://laksyvihko.fi/math.svg?latex=%3D%5Cleft%28%5Cfrac%7B1%7D%7B7%7D%7B%2C%7D%5C%20%5Cfrac%7B5%7D%7B7%7D%7B%2C%7D%5C%20%5Cfrac%7B24%7D%7B7%7D%5Cright%29" alt="=\left(\frac{1}{7}{,}\ \frac{5}{7}{,}\ \frac{24}{7}\right)"/></div> </div> 2025-08-05T13:50:46+03:00 https://peda.net/id/0c3c664171e 2018-05-17T12:27:17+03:00 412<br/> <span> suora a: </span> <div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%3D%5Cleft(-1%7B%2C%7D3%7B%2C%7D2%5Cright)" alt="P=\left(-1{,}3{,}2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bs%7D%3Di-j%2Bk" alt="\overline{s}=i-j+k"/><br/> suora b:  <div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(-3%7B%2C%7D2%7B%2C%7D-3%5Cright)" alt="A=\left(-3{,}2{,}-3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=B%3D%5Cleft(-1%7B%2C%7D1%7B%2C%7D0%5Cright)" alt="B=\left(-1{,}1{,}0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D%3D%5Cleft(-1-%5Cleft(-3%5Cright)%5Cright)i%2B%5Cleft(1-2%5Cright)j%2B%5Cleft(0-%5Cleft(-3%5Cright)%5Cright)k%3D2i-1j%2B3k" alt="\overline{AB}=\left(-1-\left(-3\right)\right)i+\left(1-2\right)j+\left(0-\left(-3\right)\right)k=2i-1j+3k"/> <div>suora a:<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-1%2B1t%26%5C%5C%0Ay%3D3-1t%26t%5Cin%5Cmathbb%7BR%7D%5C%5C%0Az%3D2%2B1t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=-1+1t&amp;\\&#10;y=3-1t&amp;t\in\mathbb{R}\\&#10;z=2+1t&amp;&#10;\end{cases}"/></div> </div> <div>suora b: </div> </div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-1%2B2r%26%5C%5C%0Ay%3D1-1r%26r%5Cin%5Cmathbb%7BR%7D%5C%5C%0Az%3D0%2B3r%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=-1+2r&amp;\\&#10;y=1-1r&amp;r\in\mathbb{R}\\&#10;z=0+3r&amp;&#10;\end{cases}"/></div> <div>Leikkauspisteessä koordinaatit ovat samat:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-1%2B1t%26%3D%26-1%2B2r%5C%5C%0A3-1t%26%3D%261-1r%5C%5C%0A2%2B1t%26%3D%260%2B3r%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;-1+1t&amp;=&amp;-1+2r\\&#10;3-1t&amp;=&amp;1-1r\\&#10;2+1t&amp;=&amp;0+3r&#10;\end{cases}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D2%7B%2C%7D%5C%20%5C%20%5C%20%5C%20t%3D4" alt="r=2{,}\ \ \ \ t=4"/></div> <br/> <span>V: </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D3%26%5C%5C%0Ay%3D-1%26%5C%5C%0Az%3D6%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=3&amp;\\&#10;y=-1&amp;\\&#10;z=6&amp;&#10;\end{cases}"/> <div><br/> <div> </div> <div>Kohta b) samalla tavalla. Yhtälöryhmällä ei ratkaisua =&gt; eivät leikkaa.<br/> <br/> <br/> <span class="editor underline"><b>417:</b></span><br/> <img src="http://laksyvihko.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D3-2t%26%26%5C%5C%0Ay%3D-1%2B2t%26%26t%5Cin%5Cmathbb%7BR%7D%5C%5C%0Az%3D3-t%26%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=3-2t&amp;&amp;\\&#10;y=-1+2t&amp;&amp;t\in\mathbb{R}\\&#10;z=3-t&amp;&amp;&#10;\end{cases}"/> <div><img src="http://laksyvihko.fi/math.svg?latex=A%3D%5Cleft(3%7B%2C%7D%5C%2011%7B%2C%7D%5C%209%5Cright)" alt="A=\left(3{,}\ 11{,}\ 9\right)"/><br/> <br/> </div> <span>Kohtisuoraan pisteestä A suoralle on piste A'.</span> <div>Suoran suuntavektori on <img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7Bs%7D%3D-2%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D-%5Coverline%7Bk%7D" alt="\overline{s}=-2\overline{i}+2\overline{j}-\overline{k}"/>. Tämä, ja vektori <img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7BAA%27%7D" alt="\overline{AA'}"/> ovat kohtisuorassa.</div> <div>Piste A' on <img src="http://laksyvihko.fi/math.svg?latex=A%27%5C%20%3D%5C%20%5Cleft(3-2t%7B%2C%7D%5C%20%5C%20-1%2B2t%7B%2C%7D%5C%20%5C%20%5C%203-t%5Cright)" alt="A'\ =\ \left(3-2t{,}\ \ -1+2t{,}\ \ \ 3-t\right)"/>, joten vektori</div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7BAA%27%7D%5C%20%3D%5C%20%5Cleft(3-2t-3%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(-1%2B2t-11%5Cright)%5Coverline%7Bj%7D%2B%5Cleft(3-t-9%5Cright)%5Coverline%7Bk%7D" alt="\overline{AA'}\ =\ \left(3-2t-3\right)\overline{i}+\left(-1+2t-11\right)\overline{j}+\left(3-t-9\right)\overline{k}"/></div> <img src="http://laksyvihko.fi/math.svg?latex=%3D-2t%5C%20%5Coverline%7Bi%7D%5C%20%2B%5C%20%5Cleft(2t-12%5Cright)%5Coverline%7Bj%7D%5C%20%2B%5C%20%5Cleft(-t%5C%20-6%5Cright)%5C%20%5Coverline%7Bk%7D" alt="=-2t\ \overline{i}\ +\ \left(2t-12\right)\overline{j}\ +\ \left(-t\ -6\right)\ \overline{k}"/>.<br/> <div> </div> <div>Lasketaan vektorien s ja AA' pistetulo:</div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7Bs%7D%5Ccdot%5Coverline%7BAA%27%7D%3D-2%5Cleft(-2t%5Cright)%2B2%5Cleft(2t-12%5Cright)-1%5Cleft(-t-6%5Cright)" alt="\overline{s}\cdot\overline{AA'}=-2\left(-2t\right)+2\left(2t-12\right)-1\left(-t-6\right)"/></div> <img src="http://laksyvihko.fi/math.svg?latex=%3D4t%2B4t-24%2Bt%2B6%5C%20%3D%5C%209t%5C%20-18" alt="=4t+4t-24+t+6\ =\ 9t\ -18"/><br/> <div><br/> Koska s ja AA' kohtisuorassa, pistetulo on nolla:</div> <div><img src="http://laksyvihko.fi/math.svg?latex=9t-18%3D0" alt="9t-18=0"/></div> <img src="http://laksyvihko.fi/math.svg?latex=9t%3D18" alt="9t=18"/><br/> <img src="http://laksyvihko.fi/math.svg?latex=t%3D%5Cfrac%7B18%7D%7B9%7D%3D2" alt="t=\frac{18}{9}=2"/> <div><br/> sijoitetaan t vektoriin AA':</div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Coverline%7BAA%27%7D%3D-2%5Ccdot2%5C%20%5Coverline%7Bi%7D%5C%20%2B%5C%20%5Cleft(2%5Ccdot2-12%5Cright)%5Coverline%7Bj%7D%5C%20%2B%5C%20%5Cleft(-2%5C%20-6%5Cright)%5C%20%5Coverline%7Bk%7D" alt="\overline{AA'}=-2\cdot2\ \overline{i}\ +\ \left(2\cdot2-12\right)\overline{j}\ +\ \left(-2\ -6\right)\ \overline{k}"/></div> <img src="http://laksyvihko.fi/math.svg?latex=%3D-4%5Coverline%7Bi%7D%5C%20%2B%5C%20-8%5C%20%5Coverline%7Bj%7D%5C%20-8%5Coverline%7Bk%7D" alt="=-4\overline{i}\ +\ -8\ \overline{j}\ -8\overline{k}"/> <div> </div> <div><img src="http://laksyvihko.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAA%27%7D%5Cright%7C%3D%5Csqrt%7B4%5E2%2B8%5E2%2B8%5E2%7D%3D%5Csqrt%7B16%2B64%2B64%7D%3D%5Csqrt%7B144%7D%3D12" alt="\left|\overline{AA'}\right|=\sqrt{4^2+8^2+8^2}=\sqrt{16+64+64}=\sqrt{144}=12"/></div> <div> </div> <div><span class="editor underline">V: 12 yksikön päässä.</span></div> <div> </div> <div><img src="http://laksyvihko.fi/math.svg?latex=A%27%5C%20%3D%5C%20%5Cleft(3-2%5Ccdot2%7B%2C%7D%5C%20%5C%20-1%2B2%5Ccdot2%7B%2C%7D%5C%20%5C%20%5C%203-2%5Cright)%5C%20%3D%5C%20%5Cleft(-1%7B%2C%7D%5C%203%7B%2C%7D%5C%201%5Cright)" alt="A'\ =\ \left(3-2\cdot2{,}\ \ -1+2\cdot2{,}\ \ \ 3-2\right)\ =\ \left(-1{,}\ 3{,}\ 1\right)"/> </div> </div> </div> 2025-08-05T13:50:46+03:00