https://peda.net/id/5b01459c72e 2019-05-10T08:42:10+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> https://peda.net/id/b61e72e4763 2019-05-14T13:20:50+03:00 a) ratkaistaan mikä luvun r pitää olla, jotta vektoreiden u ja v pistetulo on 0<a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/3-2-pistetulo/332/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/3-2-pistetulo/332/sieppaa-png:file/photo/e27c5c623f8f5261369b4f976b9261c90ef42e71/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> vektoreiden pistetulo on 0, kun r on joko -2/3 tai 1<br/> on siis mahdollista valita reaaliluku r siten että vektorit ovat kohtisuorassa toisiaan vastaan<br/> <br/> b) 2019-05-14T13:19:09+03:00 https://peda.net/id/8f026108763 2019-05-14T13:10:54+03:00 a)<br/> suunnikkaan sivuvektorit<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7B1%7D%7B2%7D%5Coverline%7Bu%7D%5C%20%5Cfrac%7B1%7D%7B2%7D%5Coverline%7Bv%7D%3D1%5Coverline%7B%5Ctext%7Bi%7D%7D%2B4%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="a=\frac{1}{2}\overline{u}\ \frac{1}{2}\overline{v}=1\overline{\text{i}}+4\overline{\text{j}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=b%3D-%5Cfrac%7B1%7D%7B2%7D%5Coverline%7Bu%7D%5Ccdot%5Cfrac%7B1%7D%7B2%7D%5Coverline%7Bv%7D%3D4%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="b=-\frac{1}{2}\overline{u}\cdot\frac{1}{2}\overline{v}=4\overline{\text{i}}+\overline{\text{j}}"/><br/> b)<br/> suunnikkaan kulmat<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(%5Coverline%7Ba%7D%7B%2C%7D%5C%20%5Coverline%7Bb%7D%5Cright)%3D%5Cfrac%7B%5Coverline%7Ba%7D%5Ccdot%5Coverline%7Bb%7D%7D%7B%5Cleft%7C%5Coverline%7Ba%7D%5Cright%7C%5Cleft%7C%5Coverline%7Bb%7D%5Cright%7C%7D%3D%5Cfrac%7B8%7D%7B17%7D%3D0%7B%2C%7D4705..." alt="\cos\left(\overline{a}{,}\ \overline{b}\right)=\frac{\overline{a}\cdot\overline{b}}{\left|\overline{a}\right|\left|\overline{b}\right|}=\frac{8}{17}=0{,}4705..."/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E%7B-1%7D%3D61%7B%2C%7D9275...%C2%B0%5Capprox61%7B%2C%7D9%C2%B0" alt="\cos^{-1}=61{,}9275...°\approx61{,}9°"/></div> <div>toinen kulma voidaan laskea</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D360%C2%B0-2%5Ccdot61%7B%2C%7D9%C2%B0%3D236%7B%2C%7D14497..." alt="2x=360°-2\cdot61{,}9°=236{,}14497..."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B236%7B%2C%7D14497...%C2%B0%7D%7B2%7D%3D118%7B%2C%7D0724...%C2%B0%5Capprox118%7B%2C%7D1%C2%B0" alt="x=\frac{236{,}14497...°}{2}=118{,}0724...°\approx118{,}1°"/></div> 2019-05-14T13:10:54+03:00 https://peda.net/id/7951ff78762 2019-05-14T12:55:58+03:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%5Ccdot%5Coverline%7Bb%7D%3D2%5Ccdot5%2B1%5Ccdot0%2B%5Cleft(-5%5Cright)%5Ccdot%5Cleft(-4%5Cright)%3D-10" alt="\overline{a}\cdot\overline{b}=2\cdot5+1\cdot0+\left(-5\right)\cdot\left(-4\right)=-10"/><br/> tämä kärki ei ole suorakulmainen<br/> <br/> selvitetään kolmas sivuvektori ja selvitetään sen avulla, onko kolmion muut kärjet suorakulmaisia</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%2B%5Coverline%7Bc%7D%3D%5Coverline%7Bb%7D" alt="\overline{a}+\overline{c}=\overline{b}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bc%7D%3D%5Coverline%7Bb%7D-%5Coverline%7Ba%7D%3D%5Cleft(5%5Coverline%7B%5Ctext%7Bi%7D%7D-4k%5Cright)-%5Cleft(2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D-5%5Coverline%7B%5Ctext%7Bk%7D%7D%5Cright)" alt="\overline{c}=\overline{b}-\overline{a}=\left(5\overline{\text{i}}-4k\right)-\left(2\overline{\text{i}}+\overline{\text{j}}-5\overline{\text{k}}\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Coverline%7B%5Ctext%7Bi%7D%7D-%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="3\overline{\text{i}}-\overline{\text{j}}+\overline{\text{k}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Coverline%7Ba%7D%5Ccdot%5Coverline%7Bc%7D%3D-2%5Ccdot3%2B%5Cleft(-1%5Cright)%5Ccdot%5Cleft(-1%5Cright)%2B%5Cleft(-5%5Cright)%5Ccdot1%3D0" alt="-\overline{a}\cdot\overline{c}=-2\cdot3+\left(-1\right)\cdot\left(-1\right)+\left(-5\right)\cdot1=0"/><br/> tämä kärki on suorakulmainen</div> <div>kolmio on suorakulmainen, koska sen samasta kärjestä lähtevien sivuvektorien pistetulo on 0</div> </div> 2019-05-14T12:55:58+03:00 https://peda.net/id/4a10c7d072f 2019-05-10T09:53:17+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2Br%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{u}=\overline{\text{i}}+r\overline{\text{j}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bv%7D%3D7%5Coverline%7B%5Ctext%7Bi%7D%7D%2B4%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{v}=7\overline{\text{i}}+4\overline{\text{j}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D7%5Ccdot1%2B4%5Ccdot%20r%3D0" alt="\overline{u}\cdot\overline{v}=7\cdot1+4\cdot r=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4r%3D-7%5C%20%5Cparallel%3A4" alt="4r=-7\ \parallel:4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=r%3D-%5Cfrac%7B7%7D%7B4%7D" alt="r=-\frac{7}{4}"/> 2019-05-10T09:53:17+03:00 https://peda.net/id/cda20eb272e 2019-05-10T09:42:38+03:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D1%5Ccdot%5Cleft(-6%5Cright)%2B2%5Ccdot3%3D0" alt="\overline{u}\cdot\overline{v}=1\cdot\left(-6\right)+2\cdot3=0"/></div> <div>vektorit ovat kohtisuorassa toisiaan vastaan</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D-8%5Ccdot0%7B%2C%7D5%2B4%5Ccdot3%2B%5Cleft(-0%7B%2C%7D5%5Cright)%5Ccdot6%3D5" alt="\overline{u}\cdot\overline{v}=-8\cdot0{,}5+4\cdot3+\left(-0{,}5\right)\cdot6=5"/></div> <div>vektorit eivät ole kohtisuorassa toisiaan vastaan</div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D2%5Ccdot5%2B%5Cleft(-0%7B%2C%7D3%5Cright)%5Ccdot20%3D4" alt="\overline{u}\cdot\overline{v}=2\cdot5+\left(-0{,}3\right)\cdot20=4"/></div> <div>vektorit eivät ole kohtisuorassa toisiaan vastaan </div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D10%5Ccdot%5Cleft(-1%5Cright)%2B2%5Ccdot5%2B3%5Ccdot0%3D0" alt="\overline{u}\cdot\overline{v}=10\cdot\left(-1\right)+2\cdot5+3\cdot0=0"/></div> <div>vektorit ovat kohtisuorassa toisiaan vastaan</div> 2019-05-10T09:42:38+03:00 https://peda.net/id/3431f72e72e 2019-05-10T09:38:21+03:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D3%5Ccdot1%2B4%5Ccdot3%3D15" alt="\overline{u}\cdot\overline{v}=3\cdot1+4\cdot3=15"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D-2%5Ccdot1%2B1%5Ccdot3%2B%5Cleft(-4%5Cright)%5Ccdot%5Cleft(-1%5Cright)%3D5" alt="\overline{u}\cdot\overline{v}=-2\cdot1+1\cdot3+\left(-4\right)\cdot\left(-1\right)=5"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D1%5Ccdot0%2B5%5Ccdot0%2B2%5Ccdot1%3D2" alt="\overline{u}\cdot\overline{v}=1\cdot0+5\cdot0+2\cdot1=2"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5Ccdot%5Coverline%7Bv%7D%3D3%5Ccdot0%2B2%5Ccdot0%2B0%5Ccdot3%3D0" alt="\overline{u}\cdot\overline{v}=3\cdot0+2\cdot0+0\cdot3=0"/></div> 2019-05-10T09:38:21+03:00 https://peda.net/id/7efe087072e 2019-05-10T09:33:17+03:00 kulma A<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D%5Coverline%7BAD%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B4%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{a}=\overline{AD}=\overline{\text{i}}+4\overline{\text{j}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D%5Coverline%7BAB%7D%3D9%5Coverline%7B%5Ctext%7Bi%7D%7D%2B2%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{b}=\overline{AB}=9\overline{\text{i}}+2\overline{\text{j}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%5Ccdot%5Coverline%7Bb%7D%3D9%5Ccdot1%2B4%5Ccdot2%3D17" alt="\overline{a}\cdot\overline{b}=9\cdot1+4\cdot2=17"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Ba%7D%5Cright%7C%3D%5Csqrt%7B1%5E2%2B4%5E2%7D%3D3" alt="\left|\overline{a}\right|=\sqrt{1^2+4^2}=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Bb%7D%5Cright%7C%3D%5Csqrt%7B9%5E2%2B2%5E2%7D%3D%5Csqrt%7B85%7D" alt="\left|\overline{b}\right|=\sqrt{9^2+2^2}=\sqrt{85}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(%5Coverline%7Ba%7D%7B%2C%7D%5C%20%5Coverline%7Bb%7D%5Cright)%3D%5Cfrac%7B17%7D%7B3%5Csqrt%7B85%7D%7D" alt="\cos\left(\overline{a}{,}\ \overline{b}\right)=\frac{17}{3\sqrt{85}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E%7B-1%7D%5Cleft(%5Cfrac%7B17%7D%7B3%5Csqrt%7B85%7D%7D%5Cright)%3D52%7B%2C%7D0745...%C2%B0%5Capprox52%7B%2C%7D1%C2%B0" alt="\cos^{-1}\left(\frac{17}{3\sqrt{85}}\right)=52{,}0745...°\approx52{,}1°"/></div> <div> </div> <div>kulma C</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bc%7D%3D%5Coverline%7BCB%7D%3D6%5Coverline%7B%5Ctext%7Bi%7D%7D%2B0%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{c}=\overline{CB}=6\overline{\text{i}}+0\overline{\text{j}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bd%7D%3D%5Coverline%7BCD%7D%3D-2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B2%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="\overline{d}=\overline{CD}=-2\overline{\text{i}}+2\overline{\text{j}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bc%7D%5Ccdot%5Coverline%7Bd%7D%3D6%5Ccdot%5Cleft(-2%5Cright)%2B0%5Ccdot2%3D-12" alt="\overline{c}\cdot\overline{d}=6\cdot\left(-2\right)+0\cdot2=-12"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Bc%7D%5Cright%7C%3D%5Csqrt%7B6%5E2%2B0%5E2%7D%3D6" alt="\left|\overline{c}\right|=\sqrt{6^2+0^2}=6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cd%5Cright%7C%3D%5Csqrt%7B%5Cleft(-2%5Cright)%5E2%2B2%5E2%7D%3D2%5Csqrt%7B2%7D" alt="\left|d\right|=\sqrt{\left(-2\right)^2+2^2}=2\sqrt{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(%5Coverline%7Bc%7D%7B%2C%7D%5C%20%5Coverline%7Bd%7D%5Cright)%3D%5Cfrac%7B-12%7D%7B6%5Ccdot2%5Csqrt%7B2%7D%7D" alt="\cos\left(\overline{c}{,}\ \overline{d}\right)=\frac{-12}{6\cdot2\sqrt{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E%7B-1%7D%5Cleft(%5Cfrac%7B-12%7D%7B6%5Ccdot2%5Csqrt%7B2%7D%7D%5Cright)%3D135%C2%B0" alt="\cos^{-1}\left(\frac{-12}{6\cdot2\sqrt{2}}\right)=135°"/></div> 2019-05-10T09:33:17+03:00