https://peda.net/id/d9f42260619 2019-04-18T08:21:42+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/e40c434e7bb 2019-05-21T13:26:27+03:00 <div>4.2 Taso avaruudessa</div> <div> </div> <div>Pisteen <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(x_0%7B%2C%7D%5C%20y_0%7B%2C%7D%5C%20z_0%5Cright)" alt="A=\left(x_0{,}\ y_0{,}\ z_0\right)"/> kautta kulkevan ja vektoria<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3Da%5Coverline%7B%5Ctext%7Bi%7D%7D%2Bb%5Coverline%7B%5Ctext%7Bj%7D%7D%2Bc%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=a\overline{\text{i}}+b\overline{\text{j}}+c\overline{\text{k}}"/> vastaan kohtisuorassa olevan tason</div> <div> </div> <div>koordinaattiyhtälö on <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>normaalimuotoinen yhtälö on <img src="https://math-demo.abitti.fi/math.svg?latex=ax%2Bby%2Bcz%2Bd%3D0" alt="ax+by+cz+d=0"/></div> <div> </div> <div> </div> <div>Esim. Piste <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(-2%7B%2C%7D3%7B%2C%7D-1%5Cright)" alt="A=\left(-2{,}3{,}-1\right)"/> on tasossa, jonka normaalivektori on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D-4%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=\overline{\text{i}}+3\overline{\text{j}}-4\overline{\text{k}}"/></div> <div>Muodosta tason 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"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%5Ccdot%5Cleft(x-%5Cleft(-2%5Cright)%5Cright)%2B3%5Ccdot%5Cleft(y-3%5Cright)-4%5Ccdot%5Cleft(z-%5Cleft(-1%5Cright)%5Cright)%3D0" alt="1\cdot\left(x-\left(-2\right)\right)+3\cdot\left(y-3\right)-4\cdot\left(z-\left(-1\right)\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B2%2B3y-9-4z-4%3D0" alt="x+2+3y-9-4z-4=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B3y-4z-11%3D0" alt="x+3y-4z-11=0"/></div> <div> </div> <div>Esim. Tason yhtälö <img src="https://math-demo.abitti.fi/math.svg?latex=3x-2y%2Bz-13%3D0" alt="3x-2y+z-13=0"/></div> <div>Määritä tasolle normaalivektori</div> <div>Onko piste (1,1,1) tasossa?</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bn%7D%3D3%5Coverline%7B%5Ctext%7Bi%7D%7D-2%5Coverline%7B%5Ctext%7Bj%7D%7D%2B%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{n}=3\overline{\text{i}}-2\overline{\text{j}}+\overline{\text{k}}"/></div> <div>sijoitetaan tason yhtälöön piste</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ccdot1-2%5Ccdot1%2B1-13%3D0" alt="3\cdot1-2\cdot1+1-13=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-11%3D0" alt="-11=0"/></div> <div>epätosi</div> <div>piste ei ole tasossa</div> <div> </div> 2019-05-21T13:26:27+03:00 https://peda.net/id/7c06910472e 2019-05-10T09:04:34+03:00 <div>3.2 Pistetulo</div> <div> </div> <div>VEKTOREIDEN PISTETULON TULOS ON LUKU, EI VEKTORI</div> <div> </div> <div>ESIM</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D3%5Coverline%7B%5Ctext%7Bi%7D%7D-7%5Coverline%7B%5Ctext%7Bj%7D%7D%2B14%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{a}=3\overline{\text{i}}-7\overline{\text{j}}+14\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D-k" alt="\overline{b}=\overline{\text{i}}+3\overline{\text{j}}-k"/></div> <div>Laske pistetulo</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%5Ccdot%5Coverline%7Bb%7D%3D3%5Ccdot1%2B%5Cleft(-7%5Cright)%5Ccdot3%2B14%5Ccdot%5Cleft(-1%5Cright)%3D3-21-14%3D-32" alt="\overline{a}\cdot\overline{b}=3\cdot1+\left(-7\right)\cdot3+14\cdot\left(-1\right)=3-21-14=-32"/></div> <div>Pistetulon avulla saadaan vektoreiden välinen kulma</div> <div><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" alt="\cos\left(\overline{a}{,}\ \overline{b}\right)=\frac{\overline{a}\cdot\overline{b}}{\left|\overline{a}\right|\left|\overline{b}\right|}="/></div> <div>Koska </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos90%C2%B0%3D0" alt="\cos90°=0"/><br/> , niin vektorit ovat kohtisuorassa täsmälleen silloin kun niiden välinen pistetulo on 0</div> <div> </div> <div>ESIM</div> <div>lasketaan vektoreiden välinen kulma</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D3%5Coverline%7B%5Ctext%7Bi%7D%7D-7%5Coverline%7B%5Ctext%7Bj%7D%7D%2B14%5Coverline%7B%5Ctext%7Bk%7D%7D" alt="\overline{a}=3\overline{\text{i}}-7\overline{\text{j}}+14\overline{\text{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D%5Coverline%7B%5Ctext%7Bi%7D%7D%2B3%5Coverline%7B%5Ctext%7Bj%7D%7D-k" alt="\overline{b}=\overline{\text{i}}+3\overline{\text{j}}-k"/> </div> <div>Pistetulo laskettiin jo edellisessä esimerkissä, -32</div> <div>lasketaan vektoreiden pituudet</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Ba%7D%5Cright%7C%3D%5Csqrt%7B3%5E2%2B%5Cleft(-7%5Cright)%5E2%2B14%5E2%7D%3D%5Csqrt%7B254%7D" alt="\left|\overline{a}\right|=\sqrt{3^2+\left(-7\right)^2+14^2}=\sqrt{254}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7C%5Coverline%7Bb%7D%5Cright%7C%3D%5Csqrt%7B1%5E2%2B3%5E2%2B%5Cleft(-1%5Cright)%5E2%7D%3D%5Csqrt%7B11%7D" alt="\left|\overline{b}\right|=\sqrt{1^2+3^2+\left(-1\right)^2}=\sqrt{11}"/><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-32%7D%7B%5Csqrt%7B254%7D%5Ccdot%5Csqrt%7B11%7D%7D%3D-0%7B%2C%7D60539..." alt="\cos\left(\overline{a}{,}\ \overline{b}\right)=\frac{-32}{\sqrt{254}\cdot\sqrt{11}}=-0{,}60539..."/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E%7B-1%7D%5Cleft(-0%7B%2C%7D60539...%5Cright)%3D127%7B%2C%7D2570...%C2%B0%5Capprox127%7B%2C%7D3%C2%B0" alt="\cos^{-1}\left(-0{,}60539...\right)=127{,}2570...°\approx127{,}3°"/></div> <div> </div> 2019-05-10T09:04:34+03:00