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phy131studiof15:lectures:chapter3 [2015/07/20 13:48]
mdawber created
phy131studiof15:lectures:chapter3 [2015/08/31 09:05] (current)
mdawber [Vectors - Components]
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 {{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;</​html>​ | $v_{2x}=v_{2}\cos\theta_{2}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | | $v_{1x}=v_{1}\cos\theta_{1}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | $v_{2x}=v_{2}\cos\theta_{2}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ |
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 | $v_{Rx}=v_{1x}+v_{2x}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | | $v_{Rx}=v_{1x}+v_{2x}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ |
 | $v_{Ry}=v_{1y}+v_{2y}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | | $v_{Ry}=v_{1y}+v_{2y}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <​html>&​nbsp;&​nbsp;&​nbsp;</​html>​ |
 +
 +===== 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 =====
  
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