Question

To find the scalar projection of \({\bf{b}}\) onto \(a\). and the vector projection of \({\bf{b}}\) onto \(a\).

Short answer

The scalar projection of \({\bf{b}}\) onto \({\bf{a}}\) is \(4\).

The vector projection of \({\bf{b}}\) onto \({\bf{a}}\) is \(\langle - 1.53,3.68\rangle \).

Step-by-step solution

Concept of Scalar Projection and Vector Projection

Formula:

Write the expression for scalar projection of\({\bf{b}}\)onto\(a\).

\({{\mathop{\rm comp}\nolimits} _{\rm{a}}}{\rm{b}} = \frac{{{\rm{a}} \cdot {\rm{b}}}}{{|{\rm{a}}|}}\)

Write the expression for vector projection of\({\bf{b}}\)onto\(a\).

\(pro{j_{\rm{a}}}{\rm{b}} = \frac{{{\rm{a}} \cdot {\rm{b}}}}{{|{\rm{a}}{|^2}}}{\rm{a}}\)

Calculation of the dot product\(a \cdot b\)

Consider a general expression to find dot product between two two-dimensional vectors.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= \left\langle {{a_1},{a_2}} \right\rangle \cdot \left\langle {{b_1},{b_2}} \right\rangle \\{\rm{a}} \cdot {\rm{b}} &= {a_1}{b_1} + {a_2}{b_2}\end{aligned}\)

In above equation, substitute \( - 5\) for \({a_1},12\) for \({a_2},4\) for \({b_1}\) and \(6\)for \({b_2}\).

\(\begin{aligned}{l}a \cdot b &= ( - 5)(4) + (12)(6)\\a \cdot b &= - 20 + 72\\a \cdot b &= 52\end{aligned}\)

Calculation of the scalar projection

Consider a general expression to find magnitude of a two dimensional vector that is\(a = \left\langle {{a_1},{a_2}} \right\rangle \).

\(|{\rm{a}}| = \sqrt {a_1^2 + a_2^2} \)

In above equation, substitute \( - 5\) for \({a_1}\) and \(12\) for \({a_2}\).

\(\begin{aligned}{l}|a| &= \sqrt {{{( - 5)}^2} + {{(12)}^2}} \\|a| &= \sqrt {25 + 144} \\|a| &= \sqrt {169} \\|a| &= 13\end{aligned}\)

In formula of scalar projection , substitute\(\;52\)for \(a\;.b\) and \(13\) for v

\({{\mathop{\rm comp}\nolimits} _{\rm{a}}}{\rm{b}} = \frac{{52}}{{13}} = 4\)

Thus, the scalar projection of \({\bf{b}}\) onto \({\bf{a}}\) is \(4.\)

Calculation of the vector projection

In formula of vector projection, substitute \(52\) for \({\rm{a}} \cdot {\rm{b}},13\) for \(|{\rm{a}}|\) and \(\langle - 5,12\rangle \) for \(a\).

\(\begin{aligned}{l}{\rm{proj}}{{\rm{ }}_{\rm{a}}}{\rm{b}} &= \frac{{52}}{{{{(13)}^2}}}\langle - 5,12\rangle \\{\rm{proj}}{{\rm{ }}_{\rm{a}}}{\rm{b}} &= \frac{{52}}{{169}}\langle - 5,12\rangle \\{\rm{proj}}{{\rm{ }}_{\rm{a}}}{\rm{b}} &= (0.307)\langle - 5,12\rangle \\{\rm{proj}}{{\rm{ }}_{\rm{a}}}{\rm{b}} &= \langle - 1.53,3.68\rangle \end{aligned}\)

Thus, the vector projection of \({\bf{b}}\) onto \({\bf{a}}\) is \(\langle - 1.53,3.68\rangle \).