Find the requested angle without reversing one vector

Find the requested angle without reversing one vector

Name the vertex of the requested angle and choose both arrows from that vertex. Calculate the signed dot product before taking inverse cosine. An absolute value is appropriate for the smaller angle between lines, but can change a requested vector angle.

For two nonzero vectors, rearrange the definition before using inverse cosine:

$$\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta \quad\Longrightarrow\quad \cos\theta=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|} \quad\Longrightarrow\quad \theta=\cos^{-1}\!\left(\frac{\mathbf a\cdot\mathbf b}{|\mathbf a||\mathbf b|}\right).$$

A negative ratio gives an obtuse angle, with $-1$ giving a straight angle. A positive ratio gives an acute angle, with $1$ giving zero. A zero ratio gives a right angle. #### Distinguish a line angle from a vector angle

The source’s optional “acute angle” convention uses the absolute value:

$$\phi=\cos^{-1}\!\left(\frac{|\mathbf a\cdot\mathbf b|}{|\mathbf a||\mathbf b|}\right).$$

This returns the smaller unoriented angle between the corresponding lines, in $[0,\pi/2]$, including parallel and perpendicular limiting cases.

It does not replace a requested directed-vector angle: Example 12 has negative dot product and an obtuse answer.

Keep the exact inverse cosine until rounding; its printed radian answer 1.63 is corrected to 1.64 (3 s.f.) in the worked solution and retained unchanged in the source evidence.

Self-Practice 3 retains both degree and radian results with their printed precision.

Choose arrows at the named vertex

For $\angle ABC$, choose $\overrightarrow{BA}$ and $\overrightarrow{BC}$, both starting at $B$, or reverse both and use $\overrightarrow{AB},\overrightarrow{CB}$.

Example 13 has $\overrightarrow{BA}=(0,1,2)$, $\overrightarrow{BC}=(1,2,-3)$, dot product $-4$ and angle $118.6^\circ$.

Its source sketch only illustrates arrow choice; the teaching figure shows the calculated obtuse angle.

Self-Practice 4 contrasts both-outgoing, both-incoming and mixed arrows: (2) and (3) are valid identities; (1) is not a general identity, although it happens to agree when the angle is a right angle.