The lines
$$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$$ and
$$\overrightarrow r = \left( {2\widehat i - \widehat j} \right) + m\left( {\widehat i + \widehat j + \widehat k} \right)$$
Solution
L<sub>1</sub> = $$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$$
<br><br>= $$\widehat i\left( {1 + 2l} \right) + \widehat j\left( { - 1} \right) + \widehat k\left( l \right)$$
<br><br>L<sub>2</sub> = $$\overrightarrow r = \left( {2\widehat i - \widehat j} \right) + m\left( {\widehat i + \widehat j + \widehat k} \right)$$
<br><br>= $$\widehat i\left( {2 + m} \right) + \widehat j\left( {m - 1} \right) + \widehat k\left( { - m} \right)$$
<br><br>Equating coefficient of $\widehat i$, $\widehat j$ and $\widehat k$ of L<sub>1</sub>
and L<sub>2</sub>
<br><br>2l + 1 = m + 2 ... (1)
<br><br>–1 = –1 + m ...(2)
<br><br>l = –m ...(3)
<br><br>from (ii) m = 0
<br><br>from (iii) $l$ = 0
<br><br>These values of m and $l$ do not satisfy equation (1).
<br><br>Hence the two lines do not intersect for any values of $l$ and m.
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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