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Calculating the angle between two vectors.

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Calculate the angle between the vectors $\\vec{v_1} = \\var{vec_1}$ and $\\vec{v_2} = \\var{vec_2}$.

{geogebra_applet('https://www.geogebra.org/m/zwjrb7kn',defs)}

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To find the angle $\\theta$ between the vectors $\\vec{v_1}$ and $\\vec{v_2}$, we can do this by first finding the angles each vector makes with the $x$-axis, and then finding the difference between the angles.

Hence, if $\\vec{v_1}$ has an angle of $\\alpha_1$ with the $x$-axis and $\\vec{v_2}$ has an angle of $\\alpha_2$ with the $x$-axis (where $\\alpha_2 >\\alpha_1$), then the angle between the vectors is $\\theta=\\alpha_2 - \\alpha_1$.

Finding $\\alpha_1$:

To calculate the angle $\\alpha_1$ between the vector $\\vec{v_1}$ and the $x$-axis, it can be helpful to draw the vector with the $x$ and $y$ components also included. This allows us to view the vector and its components as a right-angled triangle, and calculate the angle using trigonometry.

{geogebra_applet('https://www.geogebra.org/m/qrasq4rv',def1)}

We can see that we have a triangle which has an adjacent edge of length $\\var{a1}$ and an opposite edge of length $\\var{b1}$. Therefore,

\\[ \\begin{split} \\tan(\\alpha_1) &\\,= \\frac{\\var{b1}}{\\var{a1}} \\\\ \\implies \\alpha_1 &\\,= \\tan^{-1}\\left(\\simplify[fractionNumbers]{{b1/a1}}\\right) \\\\ &\\,=\\var{angle1}^\\circ \\, \\var{dp1} \\end{split} \\]

Finding $\\alpha_2$:

Calculating the angle $\\alpha_2$ between the vector $\\vec{v_2}$ and the $x$-axis using the same process:

{geogebra_applet('https://www.geogebra.org/m/hajgeg9r',def2)}

We have a triangle which has an adjacent edge of length $\\var{a2}$ and an opposite edge of length $\\var{b2}$. Therefore,

\\[ \\begin{split} \\tan(\\alpha_2) &\\,= \\frac{\\var{b2}}{\\var{a2}} \\\\ \\implies \\alpha_2 &\\,= \\tan^{-1}\\left(\\simplify[fractionNumbers]{{b2/a2}}\\right) \\\\ &\\,=\\var{angle2}^\\circ \\, \\var{dp2} \\end{split} \\]

Therefore, the angle between the vectors $\\vec{v_1}$ and $\\vec{v_2}$ is

\\[ \\begin{split} \\theta &\\,= \\alpha_2 - \\alpha_1 \\\\ &\\,=\\var{theta}^\\circ \\end{split} \\]

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$\\theta= $[[0]]$^\\circ$

(Give your answer in degrees, to 2 decimal places where necessary)

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