Question

To determine the value of \(ort{h_{\bf{a}}}{\bf{b}}\) and illustrate by drawing the vectors \({\bf{a}}\), \(b\),\(pro{j_{\bf{a}}}{\bf{b}}\)and \(ort{h_{\bf{a}}}{\bf{b}}\){\({\bf{a}} = \langle 1,4\rangle \) and \({\bf{b}} = \langle 2,3\rangle \)}

Short answer

The value of \(ort{h_{\bf{a}}}{\bf{b}}\)is \(\langle 1.176, - 0.294\rangle \).

Step-by-step solution

Formula to express vector projection of one onto other

The expression for vector projection of\({\bf{b}}\)onto\(a\)is\(pro{j_{\rm{a}}}{\rm{b}} = \frac{{{\rm{a}} \cdot {\rm{b}}}}{{|{\rm{a}}{|^2}}}{\rm{a}}\).

Calculation to determine the value of \(ort{h_{\bf{a}}}{\bf{b}}\)

The vectors \({\bf{a}} = \langle 1,4\rangle \) and \({\bf{b}} = \langle 2,3\rangle \).

Find the orthogonal projection of \({\bf{b}}\) onto \({\bf{a}}\).

\(\begin{aligned}{l}{{\mathop{\rm orth}\nolimits} _{\bf{a}}}{\bf{b}} &= {\bf{b}} - {{\mathop{\rm proj}\nolimits} _{\bf{a}}}{\bf{b}}\\ &= {\bf{b}} - \frac{{{\bf{a}} \cdot {\bf{b}}}}{{|{\bf{a}}{|^2}}}{\bf{a}}\\ &= \langle 2,3\rangle - \frac{{\langle 1,4\rangle \cdot \langle 2,3\rangle }}{{{{\left. {|1,4\rangle } \right|}^2}}}\langle 1,4\rangle \\ &= \langle 2,3\rangle - \frac{{(1)(2) + (4)(3)}}{{{{\left( {\sqrt {{1^2} + {4^2}} } \right)}^2}}}\langle 1,4\rangle \end{aligned}\)

On further simplification.

\(\begin{aligned}{l}{{\mathop{\rm orth}\nolimits} _{\rm{a}}}{\bf{b}} &= \langle 2,3\rangle - \frac{{14}}{{17}}\langle 1,4\rangle \\ &= \left\langle {\frac{{34}}{{17}},\frac{{51}}{{17}}} \right\rangle - \left\langle {\frac{{14}}{{17}},\frac{{56}}{{17}}} \right\rangle \\ &= \left\langle {\frac{{20}}{{17}}, - \frac{5}{{17}}} \right\rangle \\ &= \langle 1.176, - 0.294\rangle \end{aligned}\)

Hence, the value of\(pro{j_{\bf{a}}}{\bf{b}}\)and \(ort{h_{\bf{a}}}{\bf{b}}\) is \(\langle 0.824,3.294\rangle \) and \(\langle 1.176, - 0.294\rangle \).

Draw the vectors \({\bf{a}},{\bf{b}},{{\mathop{\rm proj}\nolimits} _{\bf{a}}}{\bf{b}}\) and \(ort{h_{\bf{a}}}{\bf{b}}\)

Draw the vectors \({\bf{a}},{\bf{b}},{{\mathop{\rm proj}\nolimits} _{\bf{a}}}{\bf{b}}\) and \(ort{h_{\bf{a}}}{\bf{b}}\) as shown below in Figure 1.

From the Figure 1, the vector projection of \({\bf{b}}\) onto \({\bf{a}}\) in the direction of a vector \({\bf{a}}\).