https://peda.net/id/2ddda6aa70b 2019-05-07T13:16:29+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/50311556721 2019-05-09T08:21:35+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Coverline%7B%5Ctext%7Bi%7D%7D%2B4%5Coverline%7B%5Ctext%7Bj%7D%7D" alt="3\overline{\text{i}}+4\overline{\text{j}}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Coverline%7B%5Ctext%7Bi%7D%7D-7%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="3\overline{\text{i}}-7\overline{\text{k}}"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5Coverline%7B%5Ctext%7Bj%7D%7D-7%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="4\overline{\text{j}}-7\overline{\text{k}}"/> 2019-05-09T08:21:35+03:00 https://peda.net/id/948fab1270b 2019-05-07T14:09:28+03:00 a) tosi<br/> b) epätosi, x-akselin pisteiden x-koordinaatti voi olla mitä vain, kunhan y- ja z-koordinaatit ovat 0<br/> c) tosi, piste on sillä koordinaattiakselilla, joka ei ole nolla 2019-05-07T14:09:28+03:00 https://peda.net/id/04ed575270b 2019-05-07T14:05:27+03:00 a) 3<br/> b) 5<br/> c) 4 2019-05-07T14:05:27+03:00 https://peda.net/id/04c59bae70b 2019-05-07T14:05:26+03:00 a) 3<br/> b) 5<br/> c) 4 2019-05-07T14:05:26+03:00 https://peda.net/id/fa36899670b 2019-05-07T14:05:09+03:00 a) P=(0, 3, 1)<br/> b) P´=(0, 3, 0) 2019-05-07T14:05:09+03:00 https://peda.net/id/cd4c7eae70b 2019-05-07T14:03:53+03:00 A4<br/> B1<br/> C2<br/> D3 2019-05-07T14:03:53+03:00 https://peda.net/id/9082c42470b 2019-05-07T14:02:11+03:00 a)<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%3D2%5Coverline%7B%5Ctext%7Bi%7D%7D-%5Coverline%7B%5Ctext%7Bj%7D%7D%2B2%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{u}=2\overline{\text{i}}-\overline{\text{j}}+2\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Bu%7D%5Cright%7C%3D%5Csqrt%7B2%5E2%2B%5Cleft(-1%5Cright)%5E2%2B2%5E2%7D%3D%5Csqrt%7B9%7D%3D3" alt="\left|\overline{u}\right|=\sqrt{2^2+\left(-1\right)^2+2^2}=\sqrt{9}=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%5E0%3D%5Cfrac%7B%5Coverline%7Bu%7D%7D%7B%5Cleft%7C%5Coverline%7Bu%7D%5Cright%7C%7D%3D%5Cfrac%7B2%5Coverline%7B%5Ctext%7Bi%7D%7D-%5Coverline%7B%5Ctext%7Bj%7D%7D%2B2%5Coverline%7B%5Ctext%7Bk%7D%7D%7D%7B3%7D%3D%5Cfrac%7B2%7D%7B3%7D%5Coverline%7B%5Ctext%7Bi%7D%7D-%5Cfrac%7B1%7D%7B3%7D%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{u}^0=\frac{\overline{u}}{\left|\overline{u}\right|}=\frac{2\overline{\text{i}}-\overline{\text{j}}+2\overline{\text{k}}}{3}=\frac{2}{3}\overline{\text{i}}-\frac{1}{3}\overline{\text{j}}+\frac{2}{3}\overline{\text{k}}"/><br/> b)<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bv%7D%3D-15%5Coverline%7Bu%7D%5E0%3D-10%5Coverline%7B%5Ctext%7Bi%7D%7D%2B5%5Coverline%7B%5Ctext%7Bj%7D%7D-10%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{v}=-15\overline{u}^0=-10\overline{\text{i}}+5\overline{\text{j}}-10\overline{\text{k}}"/><br/> <br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOP%7D%3D%5Coverline%7BOA%7D%2B%5Coverline%7Bv%7D%3D%5Coverline%7BOA%7D%2B-15%5Coverline%7Bu%7D%5E0%3D-8%5Coverline%7B%5Ctext%7Bi%7D%7D%2B5%5Coverline%7B%5Ctext%7Bj%7D%7D-15%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{OP}=\overline{OA}+\overline{v}=\overline{OA}+-15\overline{u}^0=-8\overline{\text{i}}+5\overline{\text{j}}-15\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=P%3D%5Cleft(-8%7B%2C%7D%5C%205%7B%2C%7D%5C%20-15%5Cright)" alt="P=\left(-8{,}\ 5{,}\ -15\right)"/> 2019-05-07T14:02:11+03:00 https://peda.net/id/3b777ebc70b 2019-05-07T13:54:37+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBC%7D%3D%5Coverline%7BAD%7D" alt="\overline{BC}=\overline{AD}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBC%7D%3D4%5Coverline%7B%5Ctext%7Bi%7D%7D%2B2%5Coverline%7B%5Ctext%7Bj%7D%7D%2B5%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{BC}=4\overline{\text{i}}+2\overline{\text{j}}+5\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOD%7D%3D%5Coverline%7BOA%7D%2B%5Coverline%7BAD%7D%3D-2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Coverline%7B%5Ctext%7Bj%7D%7D%2B3%5Coverline%7B%5Ctext%7Bk%7D%7D%2B4%5Coverline%7B%5Ctext%7Bi%7D%7D%2B2%5Coverline%7B%5Ctext%7Bj%7D%7D%2B5%5Coverline%7B%5Ctext%7Bk%7D%7D%3D2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D%2B8%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{OD}=\overline{OA}+\overline{AD}=-2\overline{\text{i}}+\overline{\text{j}}+3\overline{\text{k}}+4\overline{\text{i}}+2\overline{\text{j}}+5\overline{\text{k}}=2\overline{\text{i}}+3\overline{\text{j}}+8\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=D%3D%5Cleft(2%7B%2C%7D%5C%203%7B%2C%7D%5C%208%5Cright)" alt="D=\left(2{,}\ 3{,}\ 8\right)"/> <br/> b)<br/> <a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/2kk/252/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/2kk/252/sieppaa-png:file/photo/aaecfd0d3e0394b7fa2ae9c421c09ebff0f2d8b3/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a> 2019-05-07T13:52:39+03:00 https://peda.net/id/74bb42be70b 2019-05-07T13:46:02+03:00 a)<br/> <a href="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/2kk/251/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/maa4p-vektorit/2kk/251/sieppaa-png:file/photo/a4e4cd272bc45358d625e28d8f163813e0a03bdb/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%7BAB%7D%3D%5Cleft(x_2-x_1%5Cright)%5Coverline%7B%5Ctext%7Bi%7D%7D%2B%5Cleft(y_2-y_1%5Cright)%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Cleft(z_2-z_1%5Cright)%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{AB}=\left(x_2-x_1\right)\overline{\text{i}}+\left(y_2-y_1\right)\overline{\text{j}}+\left(z_2-z_1\right)\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAB%7D%3D-2%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D%2B6%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{AB}=-2\overline{\text{i}}+3\overline{\text{j}}+6\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BAC%7D%3D3%5Coverline%7B%5Ctext%7Bi%7D%7D-6%5Coverline%7B%5Ctext%7Bj%7D%7D%2B2%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{AC}=3\overline{\text{i}}-6\overline{\text{j}}+2\overline{\text{k}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BBC%7D%3D5%5Coverline%7B%5Ctext%7Bi%7D%7D-9%5Coverline%7B%5Ctext%7Bj%7D%7D-4%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{BC}=5\overline{\text{i}}-9\overline{\text{j}}-4\overline{\text{k}}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAB%7D%5Cright%7C%3D%5Csqrt%7B%5Cleft(-2%5Cright)%5E2%2B3%5E2%2B6%5E2%7D%3D%5Csqrt%7B49%7D%3D7%5C%20%5Cleft(tai%5C%20-7%5Cright)" alt="\left|\overline{AB}\right|=\sqrt{\left(-2\right)^2+3^2+6^2}=\sqrt{49}=7\ \left(tai\ -7\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BAC%7D%5Cright%7C%3D%5Csqrt%7B3%5E2%2B%5Cleft(-6%5Cright)%5E2%2B2%5E2%7D%3D%5Csqrt%7B49%7D%3D7%5C%20%5Cleft(tai%5C%20-7%5Cright)" alt="\left|\overline{AC}\right|=\sqrt{3^2+\left(-6\right)^2+2^2}=\sqrt{49}=7\ \left(tai\ -7\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BBC%7D%5Cright%7C%3D%5Csqrt%7B5%5E2%2B%5Cleft(-9%5Cright)%5E2%2B%5Cleft(-4%5Cright)%5E2%7D%3D%5Csqrt%7B122%7D" alt="\left|\overline{BC}\right|=\sqrt{5^2+\left(-9\right)^2+\left(-4\right)^2}=\sqrt{122}"/></div> <div>kolmio ABC on tasakylkinen, kyljet AB ja AC ovat yhtä pitkät</div> 2019-05-07T13:39:56+03:00 https://peda.net/id/ebefb46a70b 2019-05-07T13:36:07+03:00 a) <br/> P=(4, -1, 8)<br/> Q=(4, -4, 7)<br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BOP%7D%5Cright%7C%3D%5Csqrt%7B4%5E2%2B%5Cleft(-1%5Cright)%5E2%2B8%5E2%7D%3D%5Csqrt%7B81%7D%3D9" alt="\left|\overline{OP}\right|=\sqrt{4^2+\left(-1\right)^2+8^2}=\sqrt{81}=9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7BOQ%7D%5Cright%7C%3D%5Csqrt%7B4%5E2%2B%5Cleft(-4%5Cright)%5E2%2B7%5E2%7D%3D%5Csqrt%7B79%7D" alt="\left|\overline{OQ}\right|=\sqrt{4^2+\left(-4\right)^2+7^2}=\sqrt{79}"/><br/> Q on lähempänä origoa koska vektori OQ on lyhyempi 2019-05-07T13:36:07+03:00