Question

Given u = (1, 6) and y = (4, -9), find the projection of u onto v. Round to two decimal places, if necessary,Answer

Answer 1

It is given that,

`\begin{gathered} u\text{ = }\langle\text{ 1, 6 }\rangle \\ v\text{ = }\langle\text{ 4, -9 }\rangle \\ \parallel\bar{v}\parallel\text{ = }√(\left(4\right)^2)\text{ + \lparen-9\rparen}^2 \\ \parallel\bar{v}\parallel\text{ = }√(16+81) \\ \parallel\bar{v}\parallel\text{ = }√(97) \end{gathered}`

Calculating the dot product.

`\begin{gathered} \bar{u}\bar{.v}=\text{ }\langle1,6\rangle\text{ . }\langle4,-9\rangle \\ \bar{u}\bar{.v}=\text{ 4 - 54} \\ \bar{u}\bar{.v}=\text{ 50} \end{gathered}`

Scalar projection u onto v is given as,

`\begin{gathered} Scalar\text{ projection = }\frac{\bar{u}.\bar{v}}{\parallel\bar{v}\parallel} \\ Scalar\text{ projection =}(50)/(√(97)) \\ Scalar\text{ projection = }(50)/(9.85) \\ Scalar\text{ projection = 5.076} \\ \end{gathered}`

Vector projection of u onto v is calculated as,

`Vector\text{ projection = }\frac{\bar{u}\bar{.v}}{\parallel v\parallel^2}\bar{\text{ v}}`

Therefore,

`\begin{gathered} Vector\text{ projection = }(50)/((√(97))^2)\text{ }*\bar{\text{ v}} \\ Vector\text{ projection = }(50)/((97))\text{ }*\bar{\text{ v}} \\ Vector\bar{\text{ projection = 0.5155}}\bar{\text{ v}} \end{gathered}`

Thus the required answer is,

`\begin{gathered} Scalar\text{ projection = 5.076} \\ Vector\bar{\text{ projection = 0.5155 }}\bar{.v} \end{gathered}`