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phy131studiof15:lectures:chapter3 [2015/07/20 13:48] mdawber created |
phy131studiof15:lectures:chapter3 [2015/08/31 08:47] mdawber [Vectors - Multiplication by a scalar] |
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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: | ||
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| + | $\vec{v}\,\mathrm{ms^{-1}}=v_{x}\,\mathrm{ms^{-1}}\,\hat{i}+v_{y}\,\mathrm{ms^{-1}}\,\hat{j}+v_{z}\,\mathrm{ms^{-1}}\,\hat{k}$ | ||
| + | |||
| + | or | ||
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| + | $\vec{r}\,\mathrm{m}=x\,\mathrm{m}\,\hat{i}+y\,\mathrm{m}\,\hat{j}+z\,\mathrm{m}\,\hat{k}$ | ||
| + | |||
| + | ===== Vectors and motion ===== | ||
| + | |||
| + | |||
| + | $\vec{r_{1}}=x_{1}\,\hat{i}+y_{1}\,\hat{j}+z_{1}\,\hat{k}$ | ||
| + | |||
| + | $\vec{r_{2}}=x_{2}\,\hat{i}+y_{2}\,\hat{j}+z_{2}\,\hat{k}$ | ||
| + | |||
| + | $\Delta\vec{r}=\vec{r_{2}}-\vec{r_{1}}=(x_{2}-x_{1})\,\hat{i}+(y_{2}-y_{1})\,\hat{j}+(z_{2}-z_{1})\,\hat{k}$ | ||
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| + | Average velocity: $\vec{v_{ave}}=\frac{\Delta\vec{r}}{\Delta t}$ | ||
| + | |||
| + | Instantaneous velocity: $\vec{v}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\,\hat{i}+\frac{dy}{dt}\,\hat{j}+\frac{dz}{dt}\,\hat{k}=v_{x}\,\hat{i}+v_{y}\,\hat{j}+v_{z}\,\hat{k}$ | ||
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| + | 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 ===== | ||
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| | $v_{Rx}=v_{1x}+v_{2x}$ <html>   </html> | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <html>   </html> | | | $v_{Rx}=v_{1x}+v_{2x}$ <html>   </html> | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <html>   </html> | | ||
| | $v_{Ry}=v_{1y}+v_{2y}$ <html>   </html> | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <html>   </html> | | | $v_{Ry}=v_{1y}+v_{2y}$ <html>   </html> | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <html>   </html> | | ||
| + | |||
| + | ===== 3.P.027 ===== | ||
| + | |||
| + | ===== 3.P.042 ===== | ||
| + | |||
| + | ===== 3.P.051 ===== | ||
| + | |||
| + | ===== 3.P.062 ===== | ||
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| + | |||
| + | ===== 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 ===== | ||
