Show that lines :r=(i+j-k)+ (2 i-2 j+k) and r=(4 i-3 j+2 k)+ (i-2… - Vidyadip

Question

$\text{Show that lines :}\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-2 \hat{j}+\hat{k}) \text { and } \bar{r}=(4 \hat{i}-3 \hat{j}+2 \hat{k})+\mu(\hat{i}-2 \hat{j}+2 \hat{k})$ intersect each other.

✓

Answer

Lines $\bar{r}=\bar{a}_1+\lambda_1 \bar{b}_1$ and $\bar{r}=\bar{a}_2+\lambda_2 \bar{b}_2$ intersect each other if and only if
$
\left(\bar{a}_2-\bar{a}_1\right) \cdot\left(\bar{b}_1 \times \bar{b}_2\right)=0
$
Here $\bar{a}_1=\hat{i}+\hat{j}-\hat{k}, \bar{a}_2=4 \hat{i}-3 \hat{j}+2 \hat{k}, \quad \bar{b}_1=2 \hat{i}-2 \hat{j}+\hat{k}, \bar{b}_2=\hat{i}-2 \hat{j}+2 \hat{k}$,
$
\begin{aligned}
& \bar{a}_2-\bar{a}_1=3 \hat{i}-4 \hat{j}+3 \hat{k} \\
& \left(\bar{a}_2-\bar{a}_1\right) \cdot\left(\bar{b}_1 \times \bar{b}_2\right)=\left|\begin{array}{lll}
3 & -4 & 3 \\
2 & -2 & 1 \\
1 & -2 & 2
\end{array}\right|=3(-2)+4(3)+3(-2)=-6+12-6=0
\end{aligned}
$
Given lines intersect each other.

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