Q

# I have a doubt, kindly clarify. - Three Dimensional Geometry - JEE Main-2

Let lines $l_{1} , l_{2}$ are in direction of $\vec{a}$ and $\vec{b}$ respectively which of the following pairs of $\vec{a}$ & $\vec{b}$ represents $l_{1}$ & $l_{2}$ are parallel

• Option 1)

$\hat{i}-2\hat{j}+\hat{k}, \hat{i}-\hat{j}+\hat{k}$

• Option 2)

$\hat{i}+\hat{j}-\hat{k}, 2\hat{i}+\hat{j}-2\hat{k}$

• Option 3)

$-2\hat{i}+2\hat{j}-2\hat{k}, \hat{i}-\hat{j}-\hat{k}$

• Option 4)

$\frac{2}{3}\hat{i}-\frac{2}{3}\hat{j}+\frac{2}{3}\hat{k}, -\hat{i}+\hat{j}-\hat{k}$

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As we have learned

Condition of Parallelism -

Two lines will be Parallel if

$\vec{b}= \lambda \vec{b_{1}}$     or   $\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}$

-

(D) is the only option where $\vec{b}=\lambda\vec{a}$ is followed

we can see that

$-\hat{i}+\hat{j}-\hat{k }=\frac{-3}{2}(\frac{2}{3}\hat{i}-\frac{2}{3}\hat{j}+\frac{2}{3}\hat{k})$

$\therefore$ Option D

Option 1)

$\hat{i}-2\hat{j}+\hat{k}, \hat{i}-\hat{j}+\hat{k}$

Option 2)

$\hat{i}+\hat{j}-\hat{k}, 2\hat{i}+\hat{j}-2\hat{k}$

Option 3)

$-2\hat{i}+2\hat{j}-2\hat{k}, \hat{i}-\hat{j}-\hat{k}$

Option 4)

$\frac{2}{3}\hat{i}-\frac{2}{3}\hat{j}+\frac{2}{3}\hat{k}, -\hat{i}+\hat{j}-\hat{k}$

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