Ejercicios resueltos de operaciones con vectores

$2\vec{u} - 3\vec{v} + \vec{w} = 2(3, -1, 2) - 3(1, 4, -3) + (2, 0, 5)$

$= (6, -2, 4) - (3, 12, -9) + (2, 0, 5)$

$= (6-3+2, -2-12+0, 4-(-9)+5) = (5, -14, 18)$

$|\vec{u}| = \sqrt{3^2 + (-1)^2 + 2^2} = \sqrt{9 + 1 + 4} = \sqrt{14}$

$|\vec{v}| = \sqrt{1^2 + 4^2 + (-3)^2} = \sqrt{1 + 16 + 9} = \sqrt{26}$

$|\vec{w}| = \sqrt{2^2 + 0^2 + 5^2} = \sqrt{4 + 0 + 25} = \sqrt{29}$

$\hat{u} = \frac{\vec{u}}{|\vec{u}|} = \frac{(3, -1, 2)}{\sqrt{14}} = \left(\frac{3}{\sqrt{14}}, \frac{-1}{\sqrt{14}}, \frac{2}{\sqrt{14}}\right)$

$\vec{u} + \vec{v} + \vec{w} = (3, -1, 2) + (1, 4, -3) + (2, 0, 5) = (6, 3, 4)$

$3\vec{a} - 2\vec{b} = 3(2, 3, -1) - 2(-1, 2, 4)$

$= (6, 9, -3) - (-2, 4, 8) = (6+2, 9-4, -3-8) = (8, 5, -11)$

$\vec{a} + 2\vec{c} = \vec{b} \Rightarrow 2\vec{c} = \vec{b} - \vec{a}$

$2\vec{c} = (-1, 2, 4) - (2, 3, -1) = (-3, -1, 5)$

$\vec{c} = \left(-\frac{3}{2}, -\frac{1}{2}, \frac{5}{2}\right)$

Para $t = 0$: $\vec{a} + 0\vec{b} = (2, 3, -1)$

Para $t = 1$: $\vec{a} + \vec{b} = (2, 3, -1) + (-1, 2, 4) = (1, 5, 3)$

Para $t = 2$: $\vec{a} + 2\vec{b} = (2, 3, -1) + 2(-1, 2, 4) = (0, 7, 7)$

Para $t = -1$: $\vec{a} - \vec{b} = (2, 3, -1) - (-1, 2, 4) = (3, 1, -5)$

$\vec{m} = \frac{\vec{a} + \vec{b}}{2} = \frac{(2, 3, -1) + (-1, 2, 4)}{2} = \frac{(1, 5, 3)}{2} = \left(\frac{1}{2}, \frac{5}{2}, \frac{3}{2}\right)$

$\vec{p} - \vec{q} = (1, -2, 3) - (4, 1, -1) = (-3, -3, 4)$

$|\vec{p} - \vec{q}| = \sqrt{(-3)^2 + (-3)^2 + 4^2} = \sqrt{9 + 9 + 16} = \sqrt{34}$

$\vec{q} - \vec{r} = (4, 1, -1) - (0, 3, 2) = (4, -2, -3)$

$|\vec{q} - \vec{r}| = \sqrt{4^2 + (-2)^2 + (-3)^2} = \sqrt{16 + 4 + 9} = \sqrt{29}$

$\vec{p} + \vec{q} + \vec{r} + \vec{s} = \vec{0} \Rightarrow \vec{s} = -(\vec{p} + \vec{q} + \vec{r})$

$\vec{p} + \vec{q} + \vec{r} = (1, -2, 3) + (4, 1, -1) + (0, 3, 2) = (5, 2, 4)$

$\vec{s} = (-5, -2, -4)$

$\vec{p} - \vec{r} = (1, -2, 3) - (0, 3, 2) = (1, -5, 1)$

$|\vec{p} - \vec{r}| = \sqrt{1^2 + (-5)^2 + 1^2} = \sqrt{1 + 25 + 1} = \sqrt{27} = 3\sqrt{3}$

$\vec{p} + \vec{q} = (1, -2, 3) + (4, 1, -1) = (5, -1, 2)$

$|\vec{p} + \vec{q}| = \sqrt{25 + 1 + 4} = \sqrt{30}$

$|\vec{p}| = \sqrt{1 + 4 + 9} = \sqrt{14}$, $|\vec{q}| = \sqrt{16 + 1 + 1} = \sqrt{18}$

$|\vec{p}| + |\vec{q}| = \sqrt{14} + \sqrt{18} \approx 3.74 + 4.24 = 7.98$

$|\vec{p} + \vec{q}| = \sqrt{30} \approx 5.48$

Se cumple: $5.48 \leq 7.98$ ✓

$|\vec{x}|^2 = a^2 + 4 + 1 = a^2 + 5 = 9$

$a^2 = 4 \Rightarrow a = \pm 2$

$\vec{x} = (2, 2, -1)$ y $\vec{y} = (1, b, 3)$

$\vec{x} \cdot \vec{y} = 2(1) + 2(b) + (-1)(3) = 2 + 2b - 3 = 2b - 1 = 0$

$b = \frac{1}{2}$

$|\vec{z}|^2 = 4 + 1 + c^2 = 5 + c^2 = 9$

$c^2 = 4 \Rightarrow c = 2$ (tomando $c > 0$)

(Observación: $\vec{z}$ no puede ser unitario para ningún $c$, porque $|\vec{z}|^2 = 5 + c^2 \geq 5$.)

$\vec{x} = (2, 2, -1)$, $\vec{y} = \left(1, \frac{1}{2}, 3\right)$, $\vec{z} = (2, 1, 2)$

$\vec{x} + \vec{y} - \vec{z} = (2, 2, -1) + \left(1, \frac{1}{2}, 3\right) - (2, 1, 2)$

$= \left(2+1-2, 2+\frac{1}{2}-1, -1+3-2\right) = \left(1, \frac{3}{2}, 0\right)$ ✓