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 phy131studiof15:lectures:chapter3 [2015/07/20 13:48]mdawber created phy131studiof15:lectures:chapter3 [2015/08/31 09:05]mdawber [Vectors - Components] 2015/08/31 09:05 mdawber [Vectors - Components] 2015/08/31 08:47 mdawber [Vectors - Multiplication by a scalar] 2015/07/27 16:34 mdawber [Vectors - Multiplication by a scalar] 2015/07/27 16:33 mdawber [Vectors - Components] 2015/07/27 16:21 mdawber [3.P.003] 2015/07/27 16:20 mdawber [3.P.003] 2015/07/27 16:08 mdawber [Adding and subtracting vectors] 2015/07/20 15:20 mdawber [Vectors - Components] 2015/07/20 13:50 mdawber [Adding and subtracting vectors] 2015/07/20 13:48 mdawber created Next revision Previous revision 2015/08/31 09:05 mdawber [Vectors - Components] 2015/08/31 08:47 mdawber [Vectors - Multiplication by a scalar] 2015/07/27 16:34 mdawber [Vectors - Multiplication by a scalar] 2015/07/27 16:33 mdawber [Vectors - Components] 2015/07/27 16:21 mdawber [3.P.003] 2015/07/27 16:20 mdawber [3.P.003] 2015/07/27 16:08 mdawber [Adding and subtracting vectors] 2015/07/20 15:20 mdawber [Vectors - Components] 2015/07/20 13:50 mdawber [Adding and subtracting vectors] 2015/07/20 13:48 mdawber created Line 26: Line 26: {{vectoraddsubstract.png}} {{vectoraddsubstract.png}} + ===== 3.P.003 ===== + + ===== 3.P.005 ===== + + ===== Unit Vectors ===== + + {{unitvectors.png}} + + It can be useful to express vector quantities in terms of [[http://​en.wikipedia.org/​wiki/​Unit_vector|unit vectors]]. These are dimensionless vectors of length = 1 that point along the coordinate axes. They are usually denoted with carets (hats), i.e. $(\hat{i},​\hat{j},​\hat{k})$ + + For example: + + $\vec{v}\,​\mathrm{ms^{-1}}=v_{x}\,​\mathrm{ms^{-1}}\,​\hat{i}+v_{y}\,​\mathrm{ms^{-1}}\,​\hat{j}+v_{z}\,​\mathrm{ms^{-1}}\,​\hat{k}$ + + or + + $\vec{r}\,​\mathrm{m}=x\,​\mathrm{m}\,​\hat{i}+y\,​\mathrm{m}\,​\hat{j}+z\,​\mathrm{m}\,​\hat{k}$ + + ===== Vectors and motion ===== + + + $\vec{r_{1}}=x_{1}\,​\hat{i}+y_{1}\,​\hat{j}+z_{1}\,​\hat{k}$ + + $\vec{r_{2}}=x_{2}\,​\hat{i}+y_{2}\,​\hat{j}+z_{2}\,​\hat{k}$ + + $\Delta\vec{r}=\vec{r_{2}}-\vec{r_{1}}=(x_{2}-x_{1})\,​\hat{i}+(y_{2}-y_{1})\,​\hat{j}+(z_{2}-z_{1})\,​\hat{k}$ + + Average velocity: $\vec{v_{ave}}=\frac{\Delta\vec{r}}{\Delta t}$ + + Instantaneous velocity: $\vec{v}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\,​\hat{i}+\frac{dy}{dt}\,​\hat{j}+\frac{dz}{dt}\,​\hat{k}=v_{x}\,​\hat{i}+v_{y}\,​\hat{j}+v_{z}\,​\hat{k}$ + + Average acceleration:​ $\vec{a_{ave}}=\frac{\Delta\vec{v}}{\Delta t}$ + + Instantaneous acceleration:​ $\vec{a}=\frac{d\vec{v}}{dt}=\frac{dv_x}{dt}\,​\hat{i}+\frac{dv_y}{dt}\,​\hat{j}+\frac{dv_z}{dt}\,​\hat{k}=\frac{d^{2}x}{dt^2}\,​\hat{i}+\frac{d^{2}y}{dt^2}\,​\hat{j}+\frac{d^{2}z}{dt^2}\,​\hat{k}$ ===== Vectors - Components ===== ===== Vectors - Components ===== {{vectorcomponentadd.png}} {{vectorcomponentadd.png}} + The angles $\theta_{1}$ and $\theta_{2}$ are defined with respect to the positive $x$ axis, ie. $\theta_{1}$ is negative and $\theta_{2}$ is positive. | $v_{1x}=v_{1}\cos\theta_{1}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | $v_{2x}=v_{2}\cos\theta_{2}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | | $v_{1x}=v_{1}\cos\theta_{1}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | $v_{2x}=v_{2}\cos\theta_{2}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | Line 37: Line 72: | $v_{Rx}=v_{1x}+v_{2x}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | | $v_{Rx}=v_{1x}+v_{2x}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | | $v_{Ry}=v_{1y}+v_{2y}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | | $v_{Ry}=v_{1y}+v_{2y}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <​html>&​nbsp;&​nbsp;&​nbsp;​ | + + ===== 3.P.027 ===== + + ===== 3.P.042 ===== + + ===== 3.P.051 ===== + + ===== 3.P.062 ===== + + + + + ===== Vectors - Multiplication by a scalar ===== + + Multiplication of a vector by a scalar can change the magnitude, but not the direction of the vector, ie. each component of the vector is multiplied by the scalar in the same way. + ===== 3.P.071 =====