Chapter 9, Problem 54RE

To determine

To find: The two vectors of $v$ one parallel to $w$ and other orthogonal to $w$ .

Answer to Problem 54RE

The two vectors of $v$ one parallel to $w$ and other orthogonal to $w$ are $v_1=\frac{27}{10}i+\frac{9}{10}j$ , and $v_2=-\frac{7}{10}i+\frac{21}{10}j$ respectively.

Explanation of Solution

Given information:

The first vector is

  $v=2i+3j$

The second vector is

  $w=3i+j$

Calculation:

If $v$ and $w$ are two nonzero vectors, the vector projection of $v$ onto $w$ is given by,

  $v_1=\frac{v\cdot w}{{\Vert w\Vert }^2}w$

Substitute the values in the above formula.

  $\begin{array}{c}{v}_1=\frac{v\cdot w}{{\Vert w\Vert }^2}w\\ =\frac{\left(2i+3j\right)\cdot \left(3i+j\right)}{{\left[\sqrt{{\left(3\right)}^2+{\left(1\right)}^2}\right]}^2}\left(3i+j\right)\\ =\frac{6+3}{\left[{\sqrt{10}}^2\right]}\left(3i+j\right)\\ =\frac{27}{10}i+\frac{9}{10}j\end{array}$

The decomposition of $v$ into $v_1$ and $v_2$ . Here, the $v_1$ and $v_2$ are parallel and orthogonal to $w$ respectively.

Calculate the $v_2$ .

  $\begin{array}{c}{v}_2=v-v_1\\ =\left(2i+3j\right)-\left(\frac{27}{10}i+\frac{9}{10}j\right)\\ =-\frac{7}{10}i+\frac{21}{10}j\end{array}$

Therefore, the two vectors of $v$ one parallel to $w$ and other orthogonal to $w$ are $v_1=\frac{27}{10}i+\frac{9}{10}j$ , and $v_2=-\frac{7}{10}i+\frac{21}{10}j$ respectively.