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Vectors, EXT1 EQ-Bank 21

Given the vectors  \(\textbf{a} = \textbf{i}+3\textbf{j}\)  and  \(\textbf{b} =4\textbf{i} +2\textbf{j}\), find the projection of \(\textbf{a}\) onto \(\textbf{b}\).   (2 marks)

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\(\operatorname{proj}_{\textbf{b}}\textbf{a}=\left(\dfrac{\textbf{a} \cdot \textbf{b}}{\abs{\textbf{b}}^2}\right) \textbf{b}=\left(\dfrac{10}{20}\right) \textbf{b}=\dfrac{1}{2} \displaystyle \binom{4}{2}=\binom{2}{1}\)

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\(\displaystyle \textbf{a}=\binom{1}{3}, \ \ \textbf{b}=\binom{4}{2}\)

\(\textbf{a}\cdot \textbf{b}=1 \times 4+3 \times 2=10\)

\(\abs{\textbf{b}}^2=4^2+2^2=20\)

\(\operatorname{proj}_{\textbf{b}}\textbf{a}=\left(\dfrac{\textbf{a} \cdot \textbf{b}}{\abs{\textbf{b}}^2}\right) \textbf{b}=\left(\dfrac{10}{20}\right) \textbf{b}=\dfrac{1}{2} \displaystyle \binom{4}{2}=\binom{2}{1}\)