10. Formúlublað

10.1. Vigrar

\[\begin{aligned} |\bar{a}| = a &= \sqrt{a_x^2 + a_y^2 + a_z^2} \\ \bar{a} + \bar{b} &= \begin{pmatrix}a_x+b_x \\ a_y+b_y \\ a_z+b_z \end{pmatrix} \\ c \cdot \bar{v} &= \begin{pmatrix}c \cdot v_x\\ c \cdot v_y \\ c \cdot v_z \end{pmatrix} \\ \bar{a} \cdot \bar{b} &= a_x b_x + a_y b_y \\ &= a b \cos{\phi} \\ \bar{a} \times \bar{b} &= (a_x \hat{\imath} + a_y \hat{\jmath} + a_z \hat{k}) \times (b_x \hat{\imath} + b_y \hat{\jmath} + b_z \hat{k}) \\ &= (a_y b_z - a_z b_y)\hat{\imath} + (a_z b_x - a_x b_z)\hat{\jmath} + (a_x b_y - a_y b_x)\hat{k} \\ \bar{a} \times \bar{b} &= -\bar{b} \times \bar{a} \end{aligned}\]

10.2. Hraði

\[\begin{aligned} v &= \frac{x_2-x_1}{t_2-t_1} = \frac{\Delta x}{\Delta t} \\ &= \lim_{h \to 0} \frac{x_{t+h}-x_{t}}{(t+h) - t} = \lim_{h\to 0}\frac{x_{t+h}-x_{t}}{h} = \frac{dx}{dt} \\ \end{aligned}\]

10.3. Hröðun

\[\begin{aligned} a &= \frac{v_2 - v_1}{t_2 - t_1} \\ &= \lim_{h \to 0} \frac{v_{t+h}-v_{t}}{(t+h) - t} = \lim_{h\to 0}\frac{v_{t+h}-v_{t}}{h} = \frac{dv}{dt} \\ \end{aligned}\]

10.4. Hreyfijöfnur

\[\begin{aligned} v&=v_0+a\cdot t \qquad &\text{Hraði sem fall af tíma} \\ x&=x_0 + v_0 \cdot t + \frac{1}{2} a\cdot t^2 \qquad & \text{Staðsetning sem fall af tíma} \\ v^2 &=v_0^2+2a\cdot (x-x_0) \qquad & \text{Tímaóháða jafnan} \\ x-x_0 &=\frac{1}{2}(v_0 + v)\cdot t \qquad & \text{Færsla} \end{aligned}\]

10.5. Kraftar

\[\begin{aligned} \Sigma \bar{F} &= 0 \\ \bar{F} &= m\bar{a} \\ \bar{F}_{AB} &= -\bar{F}_{BA} \\ \end{aligned}\]