Notes: `x - 2y = 7, 6x + 2y = 5` Find the angle theta (in radians and degrees) between the lines

Wednesday, June 15, 2016

$\displaystyle{x}-{2}{y}={7},{6}{x}+{2}{y}={5}$ Find the angle theta (in radians and degrees) between the lines

Given:

$\displaystyle{x}-{2}{y}={7}$

$\displaystyle{6}{x}+{2}{y}={5}$

Find the slope for line 1.

$\displaystyle{x}-{2}{y}={7}$

$\displaystyle-{2}{y}=-{x}+{7}$

$\displaystyle{y}=\frac{{1}}{{2}}{x}-\frac{{7}}{{2}}$

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

Find the slope of line 2.

$\displaystyle{6}{x}+{2}{y}={5}$

$\displaystyle{2}{y}=-{6}{x}+{5}$

$\displaystyle{y}=-{3}{x}+\frac{{5}}{{2}}$

The slope line 2 is $\displaystyle{m}_{{2}}=-{3}.$

Find the slope between two lines using the formula

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

$\displaystyle{\tan{{\left(\theta\right)}}}={\left|\frac{{{\left(-{3}\right)}-{\left(\frac{{1}}{{2}}\right)}}}{{{1}+{\left(\frac{{1}}{{2}}\right)}{\left(-{3}\right)}}}\right|}={7}$

$\displaystyle\theta={\arctan{{\left({7}\right)}}}={{81.9}}^{\circ}={1.4289}$ radians

The angle between the two lines is 81.9 degrees or 1.4289 radians.

Notes

Wednesday, June 15, 2016

$\displaystyle{x}-{2}{y}={7},{6}{x}+{2}{y}={5}$ Find the angle theta (in radians and degrees) between the lines

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