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Worked solution

Keep a negative signed component in a vector projection

Find the projection of a=(−12−2)\mathbf a=\begin{pmatrix}-1\\2\\-2\end{pmatrix} on b=(212)\mathbf b=\begin{pmatrix}2\\1\\2\end{pmatrix}.

CJC · Vectors Basic · Self-Practice 5

Answer

−49(212)-\dfrac49\begin{pmatrix}2\\1\\2\end{pmatrix}.

Compare your reasoning

Find the signed coefficient

a⋅b=(−1)2+2(1)+(−2)2=−4,b⋅b=9\mathbf a\cdot\mathbf b=(-1)2+2(1)+(-2)2=-4,\qquad\mathbf b\cdot\mathbf b=9

The dot product is negative, so the projection points opposite to b. These steps supplement the printed short answer.

Scale the target vector

proj⁡ba=a⋅bb⋅bb=−49(212)\operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b=-\frac49\begin{pmatrix}2\\1\\2\end{pmatrix}

Do not insert an absolute value into the vector projection.

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Solution notes

The printed vector answer is retained; the full explanation is authored.