Porisma

Worked solution

Separate projection length, parallel vector and perpendicular vector

Given a=(122)\mathbf a=\begin{pmatrix}1\\2\\2\end{pmatrix} and b=(212)\mathbf b=\begin{pmatrix}2\\1\\2\end{pmatrix}, find (i) the length of projection of a\mathbf a on b\mathbf b; (ii) the projection of a\mathbf a on b\mathbf b; (iii) the component of a\mathbf a perpendicular to b\mathbf b.

CJC · Vectors Basic · Example 15

(i) Length of projection

Answer

8/38/3.

Normalise the target direction before dotting

∣a⋅b^∣=∣(122)⋅122+12+22(212)∣=13∣2+2+4∣=83.\begin{aligned}|\mathbf a\cdot\widehat{\mathbf b}|&=\left|\begin{pmatrix}1\\2\\2\end{pmatrix}\cdot\frac1{\sqrt{2^2+1^2+2^2}}\begin{pmatrix}2\\1\\2\end{pmatrix}\right|\\&=\frac13|2+2+4|=\frac83.\end{aligned}

Length uses an absolute value and is a scalar. Here the signed component happens to be positive.

How did this part go?

(ii) Parallel component

Answer

89(212)\dfrac89\begin{pmatrix}2\\1\\2\end{pmatrix}.

Multiply the signed component by the unit direction

(a⋅b^)b^=83[13(212)]=89(212)(\mathbf a\cdot\widehat{\mathbf b})\widehat{\mathbf b}=\frac83\left[\frac13\begin{pmatrix}2\\1\\2\end{pmatrix}\right]=\frac89\begin{pmatrix}2\\1\\2\end{pmatrix}

Projection of a on b follows b. Reversing the roles instead gives a vector along a; these are different projections.

How did this part go?

(iii) Perpendicular component

Answer

19(−7102)\dfrac19\begin{pmatrix}-7\\10\\2\end{pmatrix}.

Subtract the parallel component

a−(a⋅b^)b^=(122)−89(212)=(−7/910/92/9)=19(−7102).\begin{aligned}\mathbf a-(\mathbf a\cdot\widehat{\mathbf b})\widehat{\mathbf b}&=\begin{pmatrix}1\\2\\2\end{pmatrix}-\frac89\begin{pmatrix}2\\1\\2\end{pmatrix}\\&=\begin{pmatrix}-7/9\\10/9\\2/9\end{pmatrix}=\frac19\begin{pmatrix}-7\\10\\2\end{pmatrix}.\end{aligned}

The residual joins the projection foot to the original vector endpoint. Its dot product with b is zero.

How did this part go?

Common mistakes

Projection of a on b and projection of b on a generally differ. In this example both lengths happen to be 8/3 because |a|=|b|=3, but the two projection vectors still point in different directions.

Solution notes

The Teacher completes all three parts. Its sketch labels (ii) along b and (iii) from the foot to the tip; it is schematic rather than a scale drawing of these coordinates.