Chapter 9, Problem 53RE

To determine

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

Answer to Problem 53RE

The two vectors of $v$ one parallel to $w$ and other orthogonal to $w$ are $v_1=\frac{4}{5}i-\frac{3}{5}j$ , and $v_2=\frac{6}{5}i+\frac{8}{5}j$ respectively.

Explanation of Solution

Given information:

The first vector is

  $v=2i+j$

The second vector is

  $w=-4i+3j$

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.

  $\begin{array}{c}{v}_1=\frac{v\cdot w}{{\Vert w\Vert }^2}w\\ =\frac{\left(2i+j\right)\cdot \left(-4i+3j\right)}{{\left[\sqrt{{\left(-4\right)}^2+{\left(-3\right)}^2}\right]}^2}\left(-4i+3j\right)\\ =\frac{-8+3}{\left[{\sqrt{25}}^2\right]}\left(-4i+3j\right)\\ =\frac{4}{5}i-\frac{3}{5}j\end{array}$

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 $v_2$ .

  $\begin{array}{c}{v}_2=v-v_1\\ =\left(2i+j\right)-\left(\frac{4}{5}i-\frac{3}{5}j\right)\\ =\frac{6}{5}i+\frac{8}{5}j\end{array}$

Therefore, the two vectors of $v$ one parallel to $w$ and other orthogonal to $w$ are $v_1=\frac{4}{5}i-\frac{3}{5}j$ , and $v_2=\frac{6}{5}i+\frac{8}{5}j$ respectively.