// Numbas version: finer_feedback_settings {"name": "LV07 Scalar product to find angles between vectors 3D", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "LV07 Scalar product to find angles between vectors 3D", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

It is given $\\bf{a} = \\var{a}$ and $\\bf{b} = \\var{b}$.

Find the angle between $\\bf{a}$ and $\\bf{b}$.

", "advice": "

To answer these questions, we want to use the equations for the scalar product. Recall:

For the vectors $ \\mathbf v = \\pmatrix{v_1 \\\\ v_2 \\\\ v_3},\\, \\mathbf w = \\pmatrix{w_1 \\\\ w_2 \\\\ w_3},$

\\[ \\begin{split} \\mathbf{v \\cdot w} &\\,= v_1 \\times w_1 + v_2 \\times w_2 + v_3 \\times w_3 \\\\\\\\ \\mathbf{v \\cdot w} &\\,= |\\mathbf v| |\\mathbf w | \\cos(\\theta), \\end{split} \\]

where $|\\mathbf v|$ and $|\\mathbf w|$ are the magnitudes of the vectors, and $\\theta$ is the angle between the vectors.

For

$$
\\bf{a} = \\var{a} \\qquad \\text{and} \\qquad \\bf{b} = \\var{b},
$$

we have

$$
\\bf{a} \\cdot \\bf{b} = \\var{adotb},
$$

and

$$
|\\bf{a}| = \\sqrt{\\var{a[0]^2 +a[1]^2 +a[2]^2}} \\approx \\var{precround(asize,2)},\\\\
|\\bf{b}| = \\sqrt{\\var{b[0]^2 +b[1]^2 +b[2]^2}}\\approx \\var{precround(bsize,2)}.
$$

So we have:
$$
\\theta = \\cos^{-1}\\left(\\frac{\\var{adotb}}{\\sqrt{\\var{a[0]^2 +a[1]^2 +a[2]^2}}\\times\\sqrt{\\var{b[0]^2 +b[1]^2 +b[2]^2}}}\\right) = \\var{precround(angle,1)}.
$$

Use this link to find some resources which will help you revise this topic.

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The angle between $\\bf{a}$ and $\\bf{b}$ is [[0]] degrees.

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