2.1 Kaksiulotteisen koordinaatiston kantavektorit

212

213

$\overline{AB}=8\overline{\text{i}}+4\overline{\text{j}}$
$\overline{BC}=-6\overline{\text{i}}+2\overline{\text{j}}$
$\overline{CA}=-2\overline{\text{i}}-6\overline{\text{j}}$
$\left|\overline{AB}\right|=\sqrt{8^2+4^2}=4\sqrt{5}\left(tai\ -4\sqrt{5}\right)$
$\left|\overline{BC}\right|=\sqrt{6^2+2^2}=2\sqrt{10}\left(tai\ 2\sqrt{10}\right)$

$\left|CA\right|=\sqrt{2^2+6^2}=2\sqrt{10}\left(tai\ 2\sqrt{10}\right)$

kolmio on tasakylkinen, koska sillä on kaksi yhtäpitkää sivua

209

$\left|\overline{a}\right|=\sqrt{6^2+\left(-8\right)^2}=10$
$\overline{a}^0=\frac{\overline{a}}{\left|\overline{a}\right|}=\frac{6\overline{\text{i}}-8\overline{\text{j}}}{10}=\frac{3}{5}\overline{\text{i}}-\frac{4}{5}\overline{\text{j}}$
$\left|\overline{b}\right|=\sqrt{\left(-5\right)^2+12^2}=13$
$\overline{b}^0=\frac{\overline{b}}{\left|\overline{b}\right|}=\frac{-5\overline{\text{i}}+12\overline{\text{j}}}{13}=-\frac{5}{13}\overline{\text{i}}+\frac{12}{13}\overline{\text{j}}$

$\overline{AP}=5\overline{a}^0-26\overline{b}^0=3\overline{\text{i}}-5\overline{\text{j}}+10\overline{\text{i}}-24\overline{\text{j}}=13\overline{\text{i}}-29\overline{\text{j}}$
$\overline{OA}+\overline{AP}=-1\overline{\text{i}}+1\overline{\text{j}}+13\overline{\text{i}}-29\overline{\text{j}}=12\overline{\text{i}}-28\overline{\text{j}}$
$P=\left(12{,}\ -28\right)$

210

$\left|\overline{a}\right|=\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{5}{12}\left(tai\ -\frac{5}{12}\right)$
$\overline{a}^0=\frac{\overline{a}}{\left|\overline{a}\right|}=\frac{\frac{1}{3}\overline{\text{i}}-\frac{1}{4}\overline{\text{j}}}{\frac{5}{12}}=\frac{4}{5}\overline{\text{i}}-\frac{3}{5}\overline{\text{j}}$

$\overline{v}=-6\overline{a}^0=-6\left(\frac{4}{5}\overline{\text{i}}-\frac{3}{5}\overline{\text{j}}\right)=-\frac{24}{5}+\frac{18}{5}=-4\ \frac{4}{5}\overline{\text{i}}+3\ \frac{3}{5}\overline{\text{j}}$

205

b)
$\overline{OB}=\overline{OA}+\overline{AB}=-3\overline{\text{i}}+7\overline{\text{j}}+8\overline{\text{i}}-2\overline{\text{j}}=5\overline{\text{i}}+5\overline{\text{j}}$
$B=\left(5{,}\ 5\right)$
c)

$\left|\overline{OB}\right|=\sqrt{5^2+5^2}=7{,}07107...cm\approx7cm$

204

a)
$\overline{a}-\overline{b}=\left(3\overline{\text{i}}-5\overline{\text{j}}\right)-\left(-2\overline{\text{i}}+7\overline{\text{j}}\right)$
$5\overline{\text{i}}-12\overline{\text{j}}$
$\left|\overline{a}-\overline{b}\right|\ voidaan\ laskea\ Pythagoraan\ lauseella$
$x^2=5^2+12^2$
$x^2=169\ \parallel\sqrt{ }$

$x=13\ \left(tai\ -13\right)$

b)
$2\overline{a}+\overline{b}=2\left(3\overline{\text{i}}-5\overline{\text{j}}\right)+\left(-2\overline{\text{i}}+7\overline{\text{j}}\right)$
$4\overline{\text{i}}-3\overline{\text{j}}$
$\left|2\overline{a}+\overline{b}\right|\ voidaan\ laskea\ Pythagoraan\ lauseella$
$x^2=4^2+3^2$
$x^2=25\ \parallel\sqrt{ }$
$x=5\ \left(tai\ -5\right)$

203

a)
$\overline{OA}=4\overline{\text{i}}+2\overline{\text{j}}$

b)
$B=\left(-5{,}\ 3\right)$

207

Piste AB jakaa kuvan janan AB suhteessa 2:1

a) muodosta vektori AB

$\overline{AB}=6-\left(-2\right)i+1-4j$

$\overline{AB}=8\overline{\text{i}}-3\overline{\text{j}}$

b) määritä piste P ensin ilman teknisiä apuvälineitä

Piste P jakaa janan AB suhteessa 2:1 eli $\overline{AP}=\frac{2}{3}\overline{AB}$

$\overline{OP}=\overline{OA}+\overline{AP}=\overline{OA}+\frac{2}{3}\overline{AB}=-2\overline{\text{i}}+4\overline{\text{j}}+\frac{16}{3}\overline{\text{i}}-2\overline{\text{j}}$
$3\frac{1}{3}\overline{\text{i}}+2\overline{\text{j}}$
$piste\ P\ =\ \left(3\ \frac{1}{3}{,}\ 2\right)$

202

a)
$\left|\overline{u}\right|^2=3^2+4^2$
$\left|\overline{u}\right|^2=25\ \parallel\sqrt{ }$
$\left|\overline{u}\right|=5\ \left(tai\ -5\right)$
$\overline{u}^0=\frac{\overline{u}}{\left|\overline{u}\right|}$
$\overline{u}^0=\frac{3\overline{\text{i}}-4\overline{\text{j}}}{5}=\frac{3}{5}\overline{\text{i}}-\frac{4}{5}\overline{\text{j}}$
b)
$\overline{v}=-15\overline{u}^0=-15\cdot\left(\frac{3}{5}\overline{\text{i}}-\frac{4}{5}\overline{\text{j}}\right)=-9\overline{\text{i}}+12\overline{\text{j}}$
c)

201

$\overline{a}=5\overline{\text{i}}-2\overline{\text{j}}$
$\overline{b}=4\overline{\text{i}}$
$\overline{c}=-2\overline{\text{j}}$
$\left|\overline{b}\right|=4$
$\left|\overline{c}\right|=2$

b)