Notes: `0.05x - 0.03y = 0.21, 0.07x + 0.02y = 0.16` Find the angle theta (in radians and degrees) between the lines

Saturday, July 4, 2009

$\displaystyle{0.05}{x}-{0.03}{y}={0.21},{0.07}{x}+{0.02}{y}={0.16}$ Find the angle theta (in radians and degrees) between the lines

Given

$\displaystyle{0.05}{x}-{0.03}{y}={0.21}$

$\displaystyle{0.07}{x}+{0.02}{y}={0.16}$

Find the slope for line 1.

$\displaystyle{0.05}{x}-{0.03}{y}={0.21}$

$\displaystyle{5}{x}-{3}{y}={21}$

$\displaystyle-{3}{y}=-{5}{x}+{21}$

$\displaystyle{y}=\frac{{5}}{{3}}{x}-{7}$

The slope of line 1 is 5/3.

Find the slope of line 2.

$\displaystyle{0.07}{x}+{0.02}{y}={0.16}$

$\displaystyle{7}{x}+{2}{y}={16}$

$\displaystyle{2}{y}=-{7}{x}+{16}$

$\displaystyle{y}=-\frac{{7}}{{2}}{x}+{8}$

The slope for line 2 is -7/2.

Find the angle between the two lines by using the formula

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

$\displaystyle{\tan{{\left(\theta\right)}}}={\left|\frac{{{\left(-\frac{{7}}{{2}}\right)}-{\left(\frac{{5}}{{3}}\right)}}}{{{1}+{\left(\frac{{5}}{{3}}\right)}{\left(-\frac{{7}}{{2}}\right)}}}\right|}$

$\displaystyle{\tan{{\left(\theta\right)}}}=\frac{{-\frac{{31}}{{6}}}}{{-\frac{{29}}{{6}}}}=\frac{{31}}{{29}}$

$\displaystyle\theta={\arctan{{\left(\frac{{31}}{{29}}\right)}}}={{46.9}}^{\circ}={0.8187}$ radians.

The angle between the given lines is 46.9 degrees or 0.8187 radians.

Notes

Saturday, July 4, 2009

$\displaystyle{0.05}{x}-{0.03}{y}={0.21},{0.07}{x}+{0.02}{y}={0.16}$ Find the angle theta (in radians and degrees) between the lines

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