Question

Find the non-parametric form of vector equation and Cartesian equations of the plane ${\displaystyle \overrightarrow{\text{r}}=(6\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})+\text{s}(-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}})+\text{t}(-5\hat{\text{i}}-4\hat{\text{j}}-5\hat{\text{k}})}$

Answer 1

${\displaystyle \overrightarrow{\text{a}}=6\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}}$

${\displaystyle \overrightarrow{\text{b}}=-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}}$

${\displaystyle \overrightarrow{\text{c}}=-5\hat{\text{i}}-4\hat{\text{j}}-5\hat{\text{k}}}$

${\displaystyle \overrightarrow{\text{b}}\times \overrightarrow{\text{c}}=\left|\begin{array}{ccc}\hat{\text{i}}& \hat{\text{j}}& \hat{\text{k}}\\ -1& 2& 1\\ -5& -4& -5\end{array}\right|}$

= ${\displaystyle \hat{\text{i}}(-10+4)-\hat{\text{j}}(5+5)+\hat{\text{k}}(4+10)}$

= ${\displaystyle -6\hat{\text{i}}-10\hat{\text{j}}+14\hat{\text{k}}}$

= ${\displaystyle -2(3\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}})}$

Non-parametric vector equation:

${\displaystyle (\overrightarrow{\text{r}}-\overrightarrow{\text{a}})\cdot (\overrightarrow{\text{b}}\times \overrightarrow{\text{c}})=\overrightarrow{0}}$

${\displaystyle [\overrightarrow{\text{r}}-(6\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})]\cdot (3\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}})=\overrightarrow{0}}$

${\displaystyle \overrightarrow{\text{r}}\cdot (3\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}})}$ = 18 – 5 – 7 = 6

${\displaystyle \overrightarrow{\text{r}}\cdot (3\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}})}$ = 6

Cartesian equation:

3x + 5y – 7z = 6

3x + 5y – 7z – 6 = 0