Notes: `5x + 2y = 16, 3x - 5y =- 1` Find the angle theta (in radians and degrees) between the lines

Thursday, September 22, 2016

$\displaystyle{5}{x}+{2}{y}={16},{3}{x}-{5}{y}=-{1}$ Find the angle theta (in radians and degrees) between the lines

Given

$\displaystyle{5}{x}+{2}{y}={16}$

$\displaystyle{3}{x}-{5}{y}=-{1}$

Find the slope of line 1.

$\displaystyle{5}{x}+{2}{y}={16}$

$\displaystyle{2}{y}=-{5}{x}+{16}$

$\displaystyle{y}=-\frac{{5}}{{2}}{x}+{8}$

The slope of line 1 is $\displaystyle{m}_{{1}}=-\frac{{5}}{{2}}.$

Find the slope of line 2.

$\displaystyle{3}{x}-{5}{y}=-{1}$

$\displaystyle-{5}{y}=-{3}{x}-{1}$

$\displaystyle{y}=\frac{{3}}{{5}}{x}+\frac{{1}}{{5}}$

The slope of line t is $\displaystyle{m}_{{2}}=\frac{{3}}{{5}}.$

Find the angle between the lines using the formula

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

$\displaystyle{\tan{{\left(\theta\right)}}}={\left|\frac{{{\left(\frac{{3}}{{5}}\right)}-{\left(-\frac{{5}}{{2}}\right)}}}{{{1}+{\left(-\frac{{5}}{{2}}\right)}{\left(\frac{{3}}{{5}}\right)}}}\right|}=\frac{{31}}{{5}}$

$\displaystyle\theta={{\tan}}^{{-{{1}}}}{\left(\frac{{31}}{{5}}\right)}={{80.8}}^{\circ}={1.4109}$ radians.

The angle between the lines is 80.8 degrees or 1.4109 radians.

Notes

Thursday, September 22, 2016

$\displaystyle{5}{x}+{2}{y}={16},{3}{x}-{5}{y}=-{1}$ Find the angle theta (in radians and degrees) between the lines

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