Notes: `x + 2y = 8, x - 2y = 2` Find the angle theta (in radians and degrees) between the lines

Tuesday, October 5, 2010

$\displaystyle{x}+{2}{y}={8},{x}-{2}{y}={2}$ Find the angle theta (in radians and degrees) between the lines

Given $\displaystyle{x}+{2}{y}={8},{x}-{2}{y}={2}$

Find the slope of line 1.

$\displaystyle{x}+{2}{y}={8}$

$\displaystyle{2}{y}=-{x}+{8}$

$\displaystyle{y}=-\frac{{1}}{{2}}{x}+{4}$

The slope of line 1 is $\displaystyle{m}_{{1}}=-\frac{{1}}{{2}}$

Find the slope of line 2.

$\displaystyle{x}-{2}{y}={2}$

$\displaystyle-{2}{y}=-{x}+{2}$

$\displaystyle{y}=\frac{{1}}{{2}}{x}-{1}$

The slope of line 2 is $\displaystyle{m}_{{2}}=\frac{{1}}{{2}}.$

Find the angle between the two lines using the formula

$\displaystyle{\tan{{\left(\theta\right)}}}={\left|\frac{{{m}_{{2}}-{m}_{{1}}}}{{{1}+{m}_{{1}},{m}_{{2}}}}\right|}$

$\displaystyle{\tan{{\left(\theta\right)}}}={\left|\frac{{{\left(\frac{{1}}{{2}}\right)}-{\left(-\frac{{1}}{{2}}\right)}}}{{{1}+{\left(-\frac{{1}}{{2}}\right)}{\left(\frac{{1}}{{2}}\right)}}}\right|}$

$\displaystyle{\mid}$ $\displaystyle{\tan{{\left(\theta\right)}}}={\left|\frac{{1}}{{{1}-\frac{{1}}{{4}}}}\right|}=\frac{{1}}{{\frac{{3}}{{4}}}}=\frac{{4}}{{3}}$

$\displaystyle\theta={\arctan{{\left(\frac{{4}}{{3}}\right)}}}={{53.1}}^{\circ}={.9273}$ radians

The angle between the two lines is 53.1 degrees or .9273 radians.

Notes

Tuesday, October 5, 2010

$\displaystyle{x}+{2}{y}={8},{x}-{2}{y}={2}$ Find the angle theta (in radians and degrees) between the lines

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