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

Question

Show that the lines $\vec{\text{r}}=(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})+\lambda(3\hat{\text{i}}-\hat{\text{j}})\ \text{and}\ \vec{\text{r}}=({4\hat{\text{i}}-\hat{\text{k}})+\mu(2\hat{\text{i}}+3\hat{\text{k}}})$intersect. Also find their point of intersection.

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Answer

General points on the lines are
$(1+3\lambda)\hat{\text{i}}+(1-\lambda)\hat{\text{j}}-\hat{\text{k}}\ \ \&\ \ ({4+2\mu)\hat{\text{i}}+(3\mu-1)\hat{\text{k}}}$
lines intersect if
$1+3\lambda=4+2\mu\ \ \ \ \dots(1);$ $1-\lambda=0\ \ \ \ \dots(2);$ $3\mu-1=-1\ \ \ \ \dots(3)$ $\text{ for some }\lambda\ \&\ \mu$
From (2) & (3) λ =1, μ = 0
substituting in equation (1)
Since, 1 + 3(1) = 4 + 2 (0) is true $\therefore$ lines interset
Point of intersection is : $4\hat{\text{i}}-\hat{\text{k}}$ or (4, 0, -1)

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