Porisma

Worked solution

Choose both angle vectors from the same vertex

A,B,CA,B,C have position vectors i+2j+3k\mathbf i+2\mathbf j+3\mathbf k, i+j+k\mathbf i+\mathbf j+\mathbf k and 2i+3j−2k2\mathbf i+3\mathbf j-2\mathbf k respectively. Find ∠ABC\angle ABC to the nearest 0.1∘0.1^\circ.

CJC · Vectors Basic · Example 13

Answer

∠ABC=118.6∘\angle ABC=118.6^\circ.

Compare your reasoning

Start both displacements at B

BA→=OA→−OB→=(123)−(111)=(012),BC→=OC→−OB→=(23−2)−(111)=(12−3).\begin{aligned}\overrightarrow{BA}&=\overrightarrow{OA}-\overrightarrow{OB}=\begin{pmatrix}1\\2\\3\end{pmatrix}-\begin{pmatrix}1\\1\\1\end{pmatrix}=\begin{pmatrix}0\\1\\2\end{pmatrix},\\\overrightarrow{BC}&=\overrightarrow{OC}-\overrightarrow{OB}=\begin{pmatrix}2\\3\\-2\end{pmatrix}-\begin{pmatrix}1\\1\\1\end{pmatrix}=\begin{pmatrix}1\\2\\-3\end{pmatrix}.\end{aligned}

The middle letter names the angle vertex. BA and BC both point away from B, implementing the repeated Teacher guidance.

Find the two lengths

∣BA→∣=02+12+22=5,∣BC→∣=12+22+(−3)2=14|\overrightarrow{BA}|=\sqrt{0^2+1^2+2^2}=\sqrt5,\qquad|\overrightarrow{BC}|=\sqrt{1^2+2^2+(-3)^2}=\sqrt{14}

These are scalar magnitudes, not new displacement vectors.

Use the signed scalar product

BA→⋅BC→=∣BA→∣∣BC→∣cos⁡∠ABC,(012)⋅(12−3)=514cos⁡∠ABC,2−6=514cos⁡∠ABC,∠ABC=cos⁡−1 ⁣(−470)≈118.6∘.\begin{aligned}\overrightarrow{BA}\cdot\overrightarrow{BC}&=|\overrightarrow{BA}||\overrightarrow{BC}|\cos\angle ABC,\\\begin{pmatrix}0\\1\\2\end{pmatrix}\cdot\begin{pmatrix}1\\2\\-3\end{pmatrix}&=\sqrt5\sqrt{14}\cos\angle ABC,\\2-6&=\sqrt5\sqrt{14}\cos\angle ABC,\\\angle ABC&=\cos^{-1}\!\left(-\frac4{\sqrt{70}}\right)\approx118.6^\circ.\end{aligned}

Both away or both towards is valid; reversing just one direction produces the supplementary angle.

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Compare your sketch

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Two directed rays leave B towards A and C, with an obtuse included angle of 118.6 degrees. This is a two-dimensional view of the plane of the given three-dimensional displacements BA=(0,1,2), BC=(1,2,−3), not a coordinate projection or new coordinates.
Two directed rays leave B towards A and C, with an obtuse included angle of 118.6 degrees. This is a two-dimensional view of the plane of the given three-dimensional displacements BA=(0,1,2), BC=(1,2,−3), not a coordinate projection or new coordinates.
Two directed rays leave B towards A and C, with an obtuse included angle of 118.6 degrees. This is a two-dimensional view of the plane of the given three-dimensional displacements BA=(0,1,2), BC=(1,2,−3), not a coordinate projection or new coordinates.

Solution notes

Both copies print the complete calculation. The small source sketch looks acute and only illustrates arrow choice; the accessible figure shows the calculated obtuse angle.