https://peda.net/id/95f1dd3c7ba 2019-05-21T12:41:19+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/35b7143e7d2 2019-05-23T09:44:53+03:00 <a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/439/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/439/sieppaa-png:file/photo/6a6cfb1833a5629e43a6f89b468ee81ca7a8811a/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> Pisteen etäisyys tasosta on 45 2019-05-23T09:44:27+03:00 https://peda.net/id/ad5532a67d2 2019-05-23T09:40:38+03:00 a) (1, 0, 0)<br/> b) (1, 2, 0)<br/> c) (0.14, 0.71, 3.43) 2019-05-23T09:40:38+03:00 https://peda.net/id/35c349227d2 2019-05-23T09:34:59+03:00 a)<br/> <a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/435/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/435/sieppaa-png:file/photo/2499069ada0cda7b1da5a1b2fbd7be4f5cb8ed05/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> suoran ja tason leikkauspiste P=(6,-5,-9)<br/> b)<br/> <div>määritetään suoran suuntavektori</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D%3D%5Cleft(0-2%5Cright)%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Cleft(-2-%5Cleft(-3%5Cright)%5Cright)%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Cleft(6-1%5Cright)%5Coverline%7B%5Ctext%7Bk%7D%7D%3D-2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D%2B5%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{AB}=\left(0-2\right)\overline{\text{i}}+\left(-2-\left(-3\right)\right)\overline{\text{j}}+\left(6-1\right)\overline{\text{k}}=-2\overline{\text{i}}+\overline{\text{j}}+5\overline{\text{k}}"/></div> <div>suoran parametriesitys</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D2-2t%26%5C%5C%0Ay%3D-3%2Bt%26%5C%5C%0Az%3D1%2B5t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=2-2t&amp;\\&#10;y=-3+t&amp;\\&#10;z=1+5t&amp;&#10;\end{cases}"/></div> <div>tason yhtälö<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B2y-z-5%3D0" alt="x+2y-z-5=0"/></div> <div>sijoitetaan parametriesitys yhtälöön</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2-2t-6%2B2t-1-5t-5%3D0" alt="2-2t-6+2t-1-5t-5=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-5t-10%3D0" alt="-5t-10=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-5t%3D10" alt="-5t=10"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=t%3D-2" alt="t=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D2-2%5Cleft(-2%5Cright)%26%3D6%5C%5C%0Ay%3D-3%2B%5Cleft(-2%5Cright)%26%3D-5%5C%5C%0Az%3D1%2B5%5Cleft(-2%5Cright)%26%3D-9%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=2-2\left(-2\right)&amp;=6\\&#10;y=-3+\left(-2\right)&amp;=-5\\&#10;z=1+5\left(-2\right)&amp;=-9&#10;\end{cases}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%3D%5Cleft(6%7B%2C%7D-5%7B%2C%7D-9%5Cright)" alt="P=\left(6{,}-5{,}-9\right)"/></div> 2019-05-23T09:30:08+03:00 https://peda.net/id/b2a160987d1 2019-05-23T09:17:43+03:00 a)<br/> Piirretään kuva<br/> <a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/445/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/445/sieppaa-png:file/photo/5b506c28373de933649782cec86f4c0a78e7878f/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> leikkauspiste on B=(-3,1,7)<br/> kulma on 57,2<br/> b)<br/> <div>suuntavektori</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBA%7D%3D%5Cleft(-5-%5Cleft(-3%5Cright)%5Cright)%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Cleft(-2-1%5Cright)%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Cleft(11-7%5Cright)%5Coverline%7B%5Ctext%7Bk%7D%7D%3D-2%5Coverline%7B%5Ctext%7Bi%7D%7D-3%5Coverline%7B%5Ctext%7Bj%7D%7D%2B4%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{BA}=\left(-5-\left(-3\right)\right)\overline{\text{i}}+\left(-2-1\right)\overline{\text{j}}+\left(11-7\right)\overline{\text{k}}=-2\overline{\text{i}}-3\overline{\text{j}}+4\overline{\text{k}}"/></div> <div>suoran yhtälö on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-3-2t%26%5C%5C%0Ay%3D1-3t%26%5C%5C%0Az%3D7%2B4t%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=-3-2t&amp;\\&#10;y=1-3t&amp;\\&#10;z=7+4t&amp;&#10;\end{cases}"/></div> <div>leikkauspiste tason <img src="https://math-demo.abitti.fi/math.svg?latex=x-4y%2B4z-21%3D0" alt="x-4y+4z-21=0"/><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-3-2t%2B%5Cleft(-4%5Cleft(1-3t%5Cright)%5Cright)%2B4%5Cleft(7%2B4t%5Cright)-21%3D0" alt="-3-2t+\left(-4\left(1-3t\right)\right)+4\left(7+4t\right)-21=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-3-2t-4%2B12t%2B28%2B16t-21%3D0" alt="-3-2t-4+12t+28+16t-21=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=26t%3D0" alt="26t=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=t%3D0" alt="t=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D-3-2%5Ccdot0%26%5C%5C%0Ay%3D1-3%5Ccdot0%26%5C%5C%0Az%3D7%2B4%5Ccdot0%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=-3-2\cdot0&amp;\\&#10;y=1-3\cdot0&amp;\\&#10;z=7+4\cdot0&amp;&#10;\end{cases}"/><br/> leikkauspiste on<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-3%7B%2C%7D1%7B%2C%7D7%5Cright)" alt="\left(-3{,}1{,}7\right)"/></div> <div>tason normaalivektorin<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D-4%5Coverline%7B%5Ctext%7Bj%7D%7D%2B4%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=\overline{\text{i}}-4\overline{\text{j}}+4\overline{\text{k}}"/> ja suoran suuntavektorin<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBA%7D" alt="\overline{BA}"/>välinen kulma</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%5Ccdot%5Coverline%7BBA%7D%3D-2%5Ccdot1-3%5Ccdot%5Cleft(-4%5Cright)%2B4%5Ccdot4%3D26" alt="\overline{n}\cdot\overline{BA}=-2\cdot1-3\cdot\left(-4\right)+4\cdot4=26"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BBA%7D%5Cright%7C%3D%5Csqrt%7B%5Cleft(-2%5Cright)%5E2%2B%5Cleft(-3%5Cright)%5E2%2B4%5E2%7D%3D%5Csqrt%7B29%7D" alt="\left|\overline{BA}\right|=\sqrt{\left(-2\right)^2+\left(-3\right)^2+4^2}=\sqrt{29}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Bn%7D%5Cright%7C%3D%5Csqrt%7B1%5E2%2B%5Cleft(-4%5Cright)%5E2%2B4%5E2%7D%3D%5Csqrt%7B33%7D" alt="\left|\overline{n}\right|=\sqrt{1^2+\left(-4\right)^2+4^2}=\sqrt{33}"/></div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(%5Coverline%7Bn%7D%7B%2C%7D%5Coverline%7BBA%7D%5Cright)%3D%5Cfrac%7B%5Coverline%7Bn%7D%5Ccdot%5Coverline%7BBA%7D%7D%7B%5Cleft%7C%5Coverline%7Bn%7D%5Cright%7C%5Cleft%7C%5Coverline%7BBA%7D%5Cright%7C%7D%3D%5Cfrac%7B26%7D%7B%5Csqrt%7B33%7D%5Ccdot%5Csqrt%7B29%7D%7D%3D0%7B%2C%7D840460..." alt="\cos\left(\overline{n}{,}\overline{BA}\right)=\frac{\overline{n}\cdot\overline{BA}}{\left|\overline{n}\right|\left|\overline{BA}\right|}=\frac{26}{\sqrt{33}\cdot\sqrt{29}}=0{,}840460..."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E%7B-1%7D%5Cleft(0%7B%2C%7D840460...%5Cright)%3D32%7B%2C%7D8112...%C2%B0%5Capprox32%7B%2C%7D8%C2%B0" alt="\cos^{-1}\left(0{,}840460...\right)=32{,}8112...°\approx32{,}8°"/></div> <div>suoran ja tason välinen kulma<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Calpha%3D90%C2%B0-32%7B%2C%7D8%C2%B0%3D57%7B%2C%7D2%C2%B0" alt="\alpha=90°-32{,}8°=57{,}2°"/></div> 2019-05-23T08:43:31+03:00 https://peda.net/id/14e94c1e7bb 2019-05-21T14:03:36+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cleft(1-2t%5Cright)-%5Cleft(2-t%5Cright)%2B3%2Bt-2%3D0" alt="2\left(1-2t\right)-\left(2-t\right)+3+t-2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2t%2B1%3D0" alt="-2t+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=t%3D0%7B%2C%7D5" alt="t=0{,}5"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ax%3D0%26%5C%5C%0Ay%3D1%7B%2C%7D5%26%5C%5C%0Az%3D3%7B%2C%7D5%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;x=0&amp;\\&#10;y=1{,}5&amp;\\&#10;z=3{,}5&amp;&#10;\end{cases}"/> 2019-05-21T14:03:36+03:00 https://peda.net/id/e3e987ba7bb 2019-05-21T13:56:58+03:00 A3<br/> piste <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(1%7B%2C%7D0%7B%2C%7D2%5Cright)" alt="A=\left(1{,}0{,}2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D-2%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=\overline{\text{i}}-2\overline{\text{j}}+\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cleft(x-1%5Cright)-2%5Cleft(y-0%5Cright)%2B1%5Cleft(z-2%5Cright)%3D0" alt="1\left(x-1\right)-2\left(y-0\right)+1\left(z-2\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-1-2y%2Bz-2%3D0" alt="x-1-2y+z-2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-2y%2Bz-3%3D0" alt="x-2y+z-3=0"/><br/> <br/> <br/> B1<br/> piste <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(1%7B%2C%7D0%7B%2C%7D1%5Cright)" alt="A=\left(1{,}0{,}1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D%2B2%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=\overline{\text{i}}+\overline{\text{j}}+2\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cleft(x-1%5Cright)%2B1%5Cleft(y-0%5Cright)%2B2%5Cleft(z-1%5Cright)%3D0" alt="1\left(x-1\right)+1\left(y-0\right)+2\left(z-1\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-1%2By%2B2z-2%3D0" alt="x-1+y+2z-2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2By%2B2z-3%3D0" alt="x+y+2z-3=0"/><br/> <br/> <br/> C2<br/> piste <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(0%7B%2C%7D1%7B%2C%7D2%5Cright)" alt="A=\left(0{,}1{,}2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D%2B2%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=\overline{\text{i}}+\overline{\text{j}}+2\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cleft(x-0%5Cright)%2B1%5Cleft(y-1%5Cright)%2B2%5Cleft(z-2%5Cright)%3D0" alt="1\left(x-0\right)+1\left(y-1\right)+2\left(z-2\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2By-1%2B2z-4%3D0" alt="x+y-1+2z-4=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2By%2B2z-5%3D0" alt="x+y+2z-5=0"/> 2019-05-21T13:55:04+03:00 https://peda.net/id/d4b82c707bb 2019-05-21T13:48:57+03:00 a) Mikä on eräs tason normaalivektori<br/> <div>Tason yhtälö on <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%2By%2B3z-4%3D0" alt="-2x+y+3z-4=0"/></div> <div>eräs normaalivektori <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D-2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D%2B3%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=-2\overline{\text{i}}+\overline{\text{j}}+3\overline{\text{k}}"/><br/> <br/> b)<br/> <div>Ovatko pisteet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%3D%5Cleft(1%7B%2C%7D1%7B%2C%7D0%5Cright)" alt="P=\left(1{,}1{,}0\right)"/></div> <div>ja</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=Q%3D%5Cleft(1%7B%2C%7D3%7B%2C%7D1%5Cright)" alt="Q=\left(1{,}3{,}1\right)"/></div> <div>tasossa?</div> <div> </div> <div>sijoitetaan pisteet tason yhtälöön</div> <div>ensin P</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Ccdot1%2B1%5Ccdot1%2B3%5Ccdot0-4%3D0" alt="-2\cdot1+1\cdot1+3\cdot0-4=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-5%3D0" alt="-5=0"/></div> <div>epätosi, piste P ei ole tasossa</div> <div>sitten Q</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Ccdot1%2B1%5Ccdot3%2B3%5Ccdot1-4%3D0" alt="-2\cdot1+1\cdot3+3\cdot1-4=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%3D0" alt="0=0"/><!--filtered attribute: style="display: inline;"--></div> <div>tosi, piste Q on tasossa<br/> c)<br/> <a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/430/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/4ta/430/sieppaa-png:file/photo/d325f094ef72ba324035b8c48660423468e9ff85/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> <br/> Kuvasta voidaan varmistaa b-kohdan tulokset:<br/> Piste P ei ole tasossa, mutta piste Q on</div> </div> 2019-05-21T13:47:29+03:00 https://peda.net/id/1c738ca47bb 2019-05-21T13:42:20+03:00 a)<br/> <div>taso kulkee pisteen <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(2%7B%2C%7D1%7B%2C%7D0%5Cright)" alt="A=\left(2{,}1{,}0\right)"/> kautta ja jonka eräs normaalivektori on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B2%5Coverline%7B%5Ctext%7Bj%7D%7D-3%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=\overline{\text{i}}+2\overline{\text{j}}-3\overline{\text{k}}"/></div> <div>muodosta tasolle yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(x-x_0%5Cright)%2Bb%5Cleft(y-y_0%5Cright)%2Bc%5Cleft(z-z_0%5Cright)%3D0" alt="a\left(x-x_0\right)+b\left(y-y_0\right)+c\left(z-z_0\right)=0"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cleft(x-2%5Cright)%2B2%5Cleft(y-1%5Cright)-3%5Cleft(z-0%5Cright)%3D0" alt="1\left(x-2\right)+2\left(y-1\right)-3\left(z-0\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-2%2B2y-2-3z%3D0" alt="x-2+2y-2-3z=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B2y-3z-4%3D0" alt="x+2y-3z-4=0"/><br/> b)<br/> <div>taso kulkee pisteen <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(4%7B%2C%7D%5C%20-2%7B%2C%7D%5C%20-3%5Cright)" alt="A=\left(4{,}\ -2{,}\ -3\right)"/> kautta ja jonka eräs normaalivektori on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D2%5Coverline%7B%5Ctext%7Bi%7D%7D-7%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=2\overline{\text{i}}-7\overline{\text{j}}+\overline{\text{k}}"/></div> <div>muodosta tasolle yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(x-x_0%5Cright)%2Bb%5Cleft(y-y_0%5Cright)%2Bc%5Cleft(z-z_0%5Cright)%3D0" alt="a\left(x-x_0\right)+b\left(y-y_0\right)+c\left(z-z_0\right)=0"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cleft(x-4%5Cright)-7%5Cleft(y-%5Cleft(-2%5Cright)%5Cright)%2B%5Cleft(z-%5Cleft(-3%5Cright)%5Cright)%3D0" alt="2\left(x-4\right)-7\left(y-\left(-2\right)\right)+\left(z-\left(-3\right)\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x-8-7y%2B14%2Bz%2B3%3D0" alt="2x-8-7y+14+z+3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x-7y%2Bz%2B9%3D0" alt="2x-7y+z+9=0"/></div> </div> 2019-05-21T13:42:20+03:00