Find the shortest distance between the lines I_1 and l_2 r=(- - -k)+ (7 -6 +k) and r=(3 +5 +7 k)+ ( -2 +k)where and are parameters.

Given that equation of lines are r =(- - -k)+ (7 -6 +k) (i) r =(3 +5 +7 k)+ ( -2 +k) (ii) The given lines are non-parallel lines as vectors 7 -6 +k and -2 +k are not parallel. There is a unique line segment P Q ( P lying on line (i) and Q on the other line (i i) ), which is at right angles to both the lines P Q is the shortest distance between the lines. Hence, the shortest possible distance between the lines =P Q. Let the position vector of the point P lying on the line r=(- - -k)+ (7 -6 +k) where ' ' is a scalar, is (7 -1) -(6 +1) +( -1) k, for some and the position vector of the point Q lying on the line r=(3 +5 +7 k)+ ( -2 +k) where ' ^ is a scalar, is ( +3) +(-2 +5) +( +7) k, for some . Now, the vector P Q=O Q-O P=( +3-7 +1) +(-2 +5+6 +1) +( +7- +1) k i.e., P Q=( -7 +4) +(-2 +6 +6) +( - +8) k; (where 'O ' is the origin), is perpendicular to both the lines, so the vector P Q is perpendicular to both the vectors 7 -6 +k and -2 +k. ( -7 +4) 7+(-2 +6 +6) (-6)+( - +8) 1=0 ( -7 +4) 1+(-2 +6 +6) (-2)+( - +8) 1=0 2 0 - 8 6 = 0 1 0 - 4 3 = 0 6 -20 =0 3 -10 =0 On solving the above equations, we get = =0 So, the position vector of the points P and Q are - - -k and 3 +5 +7 k respectively. P Q=4 +6 +8 k and |P Q|=4^2+6^2+8^2=116=2 29 units.

Find the shortest distance between the lines I_1 and l_2 r=(- - -…

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Question

Find the shortest distance between the lines $I_1$ and $l_2$
$\vec{r}=(-\hat{\imath}-\hat{\jmath}-\hat{k})+\lambda(7 \hat{\imath}-6 \hat{\jmath}+\hat{k})$ and $\vec{r}=(3 \hat{\imath}+5 \hat{\jmath}+7 \hat{k})+\mu(\hat{\imath}-2 \hat{\jmath}+\hat{k})$where $\lambda$ and $\mu$ are parameters.

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Answer

Given that equation of lines are
$\vec{r} =(-\hat{\imath}-\hat{\jmath}-\hat{k})+\lambda(7 \hat{\imath}-6 \hat{\jmath}+\hat{k})\ldots(i)$
$\vec{r} =(3 \hat{\imath}+5 \hat{\jmath}+7 \hat{k})+\mu(\hat{\imath}-2 \hat{\jmath}+\hat{k})\ldots(ii)$
The given lines are non-parallel lines as vectors $7 \hat{\imath}-6 \hat{\jmath}+\hat{k}$ and $\hat{\imath}-2 \hat{\jmath}+\hat{k}$ are not parallel.
There is a unique line segment $P Q\ ( P$ lying on line $(i)$ and $Q$ on the other line $(i i) ),$ which is at right angles to both the lines $P Q$ is the shortest distance between the lines.
Hence, the shortest possible distance between the lines $=P Q$.
Let the position vector of the point $P$ lying on the line $r=(-\hat{\imath}-\hat{\jmath}-\hat{k})+\lambda(7 \hat{\imath}-6 \hat{\jmath}+\hat{k})$ where ' $\lambda$ ' is a scalar, is $(7 \lambda-1) \hat{\imath}-(6 \lambda+1) \hat{\jmath}+(\lambda-1) \hat{k}$, for some $\lambda$ and the position vector of the point $Q$ lying on the line $r=(3 \hat{\imath}+5 \hat{\jmath}+7 \hat{k})+\mu(\hat{\imath}-2 \hat{\jmath}+\hat{k})$ where $'\mu\ ^{\prime}$ is a scalar, is $(\mu+3) \hat{\imath}+(-2 \mu+5) \hat{\jmath}+(\mu+7) \hat{k}$, for some $\mu$.
Now, the vector
$\overline{P Q}=\overline{O Q}-\overline{O P}=(\mu+3-7 \lambda+1) \hat{\imath}+(-2 \mu+5+6 \lambda+1) \hat{\jmath}+(\mu+7-\lambda+1) \hat{k}$
i.e., $\overline{P Q}=(\mu-7 \lambda+4) \hat{\imath}+(-2 \mu+6 \lambda+6) \hat{\jmath}+(\mu-\lambda+8) \hat{k}; ($where $'O\ '$ is the origin$),$ is perpendicular to both the lines, so the vector $\overline{P Q}$ is perpendicular to both the vectors $7 \hat{\imath}-6 \hat{\jmath}+\hat{k}$ and $\hat{\imath}-2 \hat{\jmath}+\hat{k}$.
$\Rightarrow(\mu-7 \lambda+4) \cdot 7+(-2 \mu+6 \lambda+6) \cdot(-6)+(\mu-\lambda+8) \cdot 1=0$
$\ (\mu-7 \lambda+4) \cdot 1+(-2 \mu+6 \lambda+6) \cdot(-2)+(\mu-\lambda+8) \cdot 1=0$
$\Rightarrow 2 0 \mu- 8 6 \lambda= 0$
$\Rightarrow 1 0 \mu- 4 3 \lambda= 0 \ 6 \mu-20 \lambda=0 $
$\Rightarrow 3 \mu-10 \lambda=0$
On solving the above equations, we get $\mu=\lambda=0$
So, the position vector of the points $P$ and $Q$ are $-\hat{\imath}-\hat{\jmath}-\hat{k}$ and $3 \hat{\imath}+5 \hat{\jmath}+7 \hat{k}$ respectively.
$\overline{P Q}=4 \hat{\imath}+6 \hat{\jmath}+8 \hat{k}$ and $|\overline{P Q}|=\sqrt{4^2+6^2+8^2}=\sqrt{116}=2 \sqrt{29}$ units.

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