Question

In Problems 53 and 54, decompose v into two vectors, one parallel to w and the other orthogonal to w.

$v=2i+j;w=-4i+3j$

Short answer

The two vectors are${\mathrm{v}}_1=\frac{4}{5}\mathrm{i}-\frac{3}{5}\mathrm{j}\mathrm{and}{\mathrm{v}}_2=\frac{6}{5}\mathrm{i}+\frac{8}{5}\mathrm{j}.$

Step-by-step solution

Step 1. Given Information 

The given vectors are$v=2i+jandw=-4i+3j.$

Step 2. Finding the vector parallel to w

Let${\mathrm{v}}_1$be the vector parallel to w.To find ${\mathrm{v}}_1$, we will use formula $v_1=\frac{\mathrm{v}·\mathrm{w}}{{\left|\left|w\right|\right|}^2}w.$

First, find the dot product

role="math" localid="1647003393909" $$\begin{gathered} \mathrm{v}·\mathrm{w}=\left(2\mathrm{i}+\mathrm{j}\right)·\left(-4\mathrm{i}+3\mathrm{j}\right) \\ \mathrm{v}·\mathrm{w}=-8+3 \\ \mathrm{v}·\mathrm{w}=-5 \end{gathered}$$

Now, let's find ${\left|\left|w\right|\right|}^2$

$$\begin{gathered} \left|\left|w\right|\right|=\sqrt{{\left(-4\right)}^2+3^2} \\ \left|\left|w\right|\right|=\sqrt{25} \\ \left|\left|w\right|\right|=5 \\ {\left|\left|w\right|\right|}^2=25 \end{gathered}$$

Put all the values in the formula

$$\begin{gathered} v_1=\frac{-5}{25}\left(-4i+3j\right) \\ {v}_1=\frac{20}{25}i-\frac{15}{25}j \\ {v}_1=\frac{4}{5}i-\frac{3}{5}j \end{gathered}$$

Step 3. Finding the vector orthogonal to w

Let $v_2$ be the vector orthogonal to w.To find $v_2$we will use the formula $v_2=v-v_1.$

So,

$$\begin{gathered} v_2=\left(2i+j\right)-\left(\frac{4}{5}i-\frac{3}{5}j\right) \\ {v}_2=\frac{6}{5}i+\frac{8}{5}j \end{gathered}$$