Question

Find all the vectors in ${\mathbf{R}}^{\mathbf{4}}$that are perpendicular to the three vectors

$\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ 9\\ 9\\ 7\end{array}\right]$

See exercise 36.

Short answer

The vector is,

${\mathrm{\nu }}_4\left(\mathrm{t}\right)=4\mathrm{t}$

$\overrightarrow{\mathrm{\nu }}=\left[\begin{array}{c}-\mathrm{t}\\ 6\mathrm{t}\\ -9\mathrm{t}\\ 4\mathrm{t}\end{array}\right]$

Step-by-step solution

Concept introduction

Two vectors are perpendicular to each other if their dot vector is zero,

Let us suppose that $\overrightarrow{\mathrm{x}}$and $\overrightarrow{\mathrm{y}}$are two perpendicular then these two vectors are said to be perpendicular, if

$\overrightarrow{\mathrm{x}}.\overrightarrow{\mathrm{y}}=0$

Given Data

The three vectors are given by,

$\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right],\left[\begin{array}{c}1\\ 9\\ 9\\ 7\end{array}\right]$

Find all the vectors that are perpendicular to the given three vectors

Let us assume that the,

$\overrightarrow{\mathrm{v}}=\left[\begin{array}{c}{\mathrm{v}}_1\\ {\mathrm{v}}_2\\ {\mathrm{v}}_3\\ {\mathrm{v}}_4\end{array}\right]$

The dot product ofdata-custom-editor="chemistry" $\overrightarrow{\mathrm{v}}$ vector with the given three vectors should be equal to zero,

$\begin{array}{rcl}\left(1\right){\mathrm{v}}_1+\left(1\right){\mathrm{v}}_2+\left(1\right){\mathrm{v}}_3+\left(1\right){\mathrm{v}}_4& =& 0\\ {\mathrm{v}}_1+{\mathrm{v}}_2+{\mathrm{v}}_3+{\mathrm{v}}_4& =& 0\end{array}$

The dot product ofdata-custom-editor="chemistry" $\overrightarrow{\mathrm{v}}$ vector with the given second vector,

data-custom-editor="chemistry" $\begin{array}{rcl}\left(1\right){\mathrm{v}}_1+\left(2\right){\mathrm{v}}_2+\left(3\right){\mathrm{v}}_3+\left(4\right){\mathrm{v}}_4& =& 0\\ {\mathrm{v}}_1+2{\mathrm{v}}_2+3{\mathrm{v}}_3+4{\mathrm{v}}_4& =& 0\end{array}$

The dot product ofdata-custom-editor="chemistry" $\overrightarrow{\mathrm{v}}$ vector with the given third vector,

$\begin{array}{rcl}\left(1\right){\mathrm{v}}_1+\left(9\right){\mathrm{v}}_2+\left(9\right){\mathrm{v}}_3+\left(7\right){\mathrm{v}}_4& =& 0\\ 1{\mathrm{v}}_1+9{\mathrm{v}}_2+9{\mathrm{v}}_3+7{\mathrm{v}}_4& =& 0\end{array}$

Make matrix$3\times 5$ from the above three calculated equation, so that columns are the components of v and the last column is the value of dot product which is equal to 0.

$\left[\begin{array}{ccccc}1& 1& 1& 1& 0\\ 1& 2& 3& 4& 0\\ 1& 9& 9& 7& 0\end{array}\right]$

Now reduce the above matrix,

Subtraft row 1 from row 2 and row 3,

$\left[\begin{array}{ccccc}1& 1& 1& 1& 0\\ 0& 1& 2& 3& 0\\ 1& 8& 8& 6& 0\end{array}\right]$

Use operation data-custom-editor="chemistry" ${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2$and data-custom-editor="chemistry" ${\mathrm{R}}_3\to {\mathrm{R}}_3-8{\mathrm{R}}_2$

$\left[\begin{array}{ccccc}1& 0& -1& -2& 0\\ 0& 1& 2& 3& 0\\ 0& 0& -8& -18& 0\end{array}\right]$

Use operation, ${\mathrm{R}}_3\to -\frac{{\mathrm{R}}_3}{8}$

$\left[\begin{array}{ccccc}1& 0& -1& -2& 0\\ 0& 1& 2& 3& 0\\ 0& 0& 1& \frac{9}{4}& 0\end{array}\right]$

Use operation, ${\mathrm{R}}_1\to {\mathrm{R}}_1+{\mathrm{R}}_3\mathrm{and}{\mathrm{R}}_2\to {\mathrm{R}}_2-2{\mathrm{R}}_3$

$\left[\begin{array}{ccccc}1& 0& 0& \frac{1}{4}& 0\\ 0& 1& 2& -\frac{3}{2}& 0\\ 0& 0& 1& \frac{9}{4}& 0\end{array}\right]$

Now from here we can write each component of the vectordata-custom-editor="chemistry" $\overrightarrow{\mathrm{v}}$ in terms of data-custom-editor="chemistry" ${\mathrm{v}}_4$as follows,

$$\begin{gathered} {\mathrm{v}}_1+\frac{1}{4}{\mathrm{v}}_4=0 \\ {\mathrm{v}}_1=-\frac{1}{4}{\mathrm{v}}_4 \end{gathered}$$

Similarly will have,

$$\begin{gathered} {\mathrm{v}}_2=\frac{3}{2}{\mathrm{v}}_4 \\ {\mathrm{v}}_3=-\frac{9}{4}{\mathrm{v}}_4 \end{gathered}$$

Let us assume that for arbitary data-custom-editor="chemistry" ${\mathrm{v}}_4=4\mathrm{t}$(for real number t), we have the solution as,

$\overrightarrow{\mathrm{v}}=\left[\begin{array}{c}-\mathrm{t}\\ 6\mathrm{t}\\ -9\mathrm{t}\\ 4\mathrm{t}\end{array}\right]$