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Vectors: lines and perpendicular feet

Choose the angle between two lines

Read which angle is asked for before using a dot product. Reversing a direction changes the directed-vector angle but preserves the smaller angle between the lines.

  1. Extract two nonzero directions

    Use the vectors multiplying the independent line parameters. The fixed position vectors locate the lines; they do not determine the angle.

    m1≠0\mathbf m_1\ne\mathbf0

    m2≠0\mathbf m_2\ne\mathbf0

    Intersecting or skewAn intersection is unnecessary

    Translate the direction vectors to a common starting point to compare them. Skew lines still have an angle between their directions; a nonzero angle alone does not prove the lines intersect.

    A zero direction does not define a line. A zero component is allowed. Keep separate parameters for the two line equations.

  2. Choose the requested angle

    For the smaller line angle, take the absolute value of the dot product. For an angle between specified directed vectors, retain its sign.

    c=m1⋅m2∣m1∣∣m2∣c=\dfrac{\mathbf m_1\cdot\mathbf m_2}{|\mathbf m_1||\mathbf m_2|}

    Smaller line angleRemove direction reversal
    α=cos⁡−1∣c∣.\alpha=\cos^{-1}|c|.

    This gives 0≤α≤π/20\le\alpha\le\pi/2. The source calls this the acute angle; the parallel and perpendicular boundaries are 00 and π/2\pi/2.

    Specified vector angleKeep the signed dot product
    θ=cos⁡−1c.\theta=\cos^{-1}c.

    This gives 0≤θ≤π0\le\theta\le\pi. For an internal angle ABC, use BA and BC, both directed from the vertex B. An absolute value could replace the required obtuse angle by its supplement.

    Inverse cosine is not a reciprocal. Set the calculator to the requested degree or radian mode, and retain full precision until the final answer.

  3. Check the angle and the geometry

    The same supporting lines can be represented using reversed or rescaled directions. Their smaller angle must stay unchanged. In radians use π − θ for the supplement; in degrees use 180° − θ.

    α=min⁡(θ,π−θ)\alpha=\min(\theta,\pi-\theta)

    Reversed directionThe vector angle may change

    Reversing one direction replaces cc by −c-c and θ\theta by π−θ\pi-\theta. The value ∣c∣|c|, and therefore the smaller line angle, stays the same.

    Boundary casesSeparate angle from line relationship

    A zero dot product gives a right angle between directions. Parallel directions give a zero smaller angle; the lines may be distinct or coincident. Check positions separately when classifying the lines.

    Use the smaller-angle rule only when that convention is requested. Do not force an internal polygon angle or a specified vector angle to be acute.

Try a worked example

Read a case, make a prediction, then reveal the reasoning. The examples keep their stated values.

Source Example 26: use the directions

The source directions are (1,2,1)(1,2,1) and (3,2,6)(3,2,6). Predict whether their directed-vector angle is already acute before calculating.

Show reasoning

  1. 1. Find the dot product and lengths

    The fixed line points do not enter the angle calculation. The positive dot product predicts an acute directed-vector angle.

    m1⋅m2=13∣m1∣=6∣m2∣=7\begin{aligned}\mathbf m_1\cdot\mathbf m_2&=13\\|\mathbf m_1|&=\sqrt6\\|\mathbf m_2|&=7\end{aligned}

  2. 2. Calculate the smaller angle

    The direction angle is already acute, so its supplement is unnecessary. Round only the final answer to one decimal place.

    α=cos⁡−1 ⁣(1376)=40.7∘(1 d.p.)\begin{aligned}\alpha&=\cos^{-1}\!\left(\frac{13}{7\sqrt6}\right)\\&=40.7^\circ\quad\text{(1 d.p.)}\end{aligned}
Continue with the lesson question

Reverse a direction, preserve the line angle

Authored comparison using Example 26: replace the second direction by (−3,−2,−6)(-3,-2,-6). Predict what changes and what stays the same. Both lines still use unrestricted real parameters.

Show reasoning

  1. 1. Keep the sign for the vector angle

    The dot product becomes minus 13. The directed-vector angle is now obtuse, although the supporting line is unchanged.

    cos⁡θ=−1376θ≈139.3∘\begin{aligned}\cos\theta&=-\frac{13}{7\sqrt6}\\\theta&\approx139.3^\circ\end{aligned}

  2. 2. Take the supplement for the line angle

    Use the unrounded angle in a calculation, or take the absolute dot product directly. Reversing or nonzero rescaling cannot change the smaller angle between these lines.

    α=180∘−θ=40.7∘(1 d.p.)\begin{aligned}\alpha&=180^\circ-\theta\\&=40.7^\circ\quad\text{(1 d.p.)}\end{aligned}
Continue with the lesson question

Source Example 27: an internal angle can be obtuse

The source asks for the internal angle OAB. Both AO→=(−3,−11,−11)\overrightarrow{AO}=(-3,-11,-11) and AB→=(2,−1,10)\overrightarrow{AB}=(2,-1,10) start at A. Predict why an absolute dot product would answer a different question.

Show reasoning

  1. 1. Use both vectors from the vertex

    The signed dot product is minus 105. Their lengths are square-root 251 and square-root 105, so the internal angle is obtuse.

    cos⁡∠OAB=−105251105\cos\angle OAB=\frac{-105}{\sqrt{251}\sqrt{105}}

  2. 2. Retain the obtuse answer

    The source rounds the internal angle to the nearest degree. Taking an absolute value would instead give about 50 degrees, the smaller angle between the supporting lines.

    ∠OAB=130∘(nearest degree)\begin{aligned}\angle OAB&=130^\circ\\&\text{(nearest degree)}\end{aligned}
Continue with the lesson question

Compare the source angle constructions

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Two source constructions: when direction vectors form an acute angle theta, the line angle alpha equals theta; reversing one direction makes theta obtuse, so alpha=pi−theta. Taking the absolute dot product gives the smaller angle directly, even for skew lines. Parallel and perpendicular boundary cases give zero and 90 degrees.
Two source constructions: when direction vectors form an acute angle theta, the line angle alpha equals theta; reversing one direction makes theta obtuse, so alpha=pi−theta. Taking the absolute dot product gives the smaller angle directly, even for skew lines. Parallel and perpendicular boundary cases give zero and 90 degrees.
Two source constructions: when direction vectors form an acute angle theta, the line angle alpha equals theta; reversing one direction makes theta obtuse, so alpha=pi−theta. Taking the absolute dot product gives the smaller angle directly, even for skew lines. Parallel and perpendicular boundary cases give zero and 90 degrees.
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