Let a¯=i¯+j¯, b¯=2i¯-k¯. Then the point of intersecton of the lines r¯×a¯=b¯×a¯ and r¯×b¯=a¯×b¯ is

r×a=b×a=-(a×b)r×b=a×b ⇒ r×a+r×b=0 ⇒ r×(a+b)=0 ⇒ r || a+b ⇒ r=a+b=3i+j-k

Let a¯=i¯+j¯, b¯=2i¯-k¯. Then the point of intersecton of the lines r¯×a¯=b¯×a¯ and r¯×b¯=a¯×b¯ is

r×a=b×a=-(a×b)r×b=a×b ⇒ r×a+r×b=0 ⇒ r×(a+b)=0 ⇒ r || a+b ⇒ r=a+b=3i+j-k

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Let $\bar{a} = \bar{i} + \bar{j}, \bar{b} = 2 \bar{i} - \bar{k}$ . Then the point of intersecton of the lines $\bar{r} \times \bar{a} = \bar{b} \times \bar{a}$ and
$\bar{r} \times \bar{b} = \bar{a} \times \bar{b}$ is

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$3 \bar{i} + \bar{j} - \bar{k}$

$3 \bar{i} - \bar{j} - \bar{k}$

$3 \bar{i} - 3 \bar{j} - \bar{k}$

$3 \bar{i} + 3 \bar{j} + \bar{k}$

answer is A.

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Detailed Solution

$\begin{array}{rcl} r \times a & = & b \times a = - (a \times b) \\ r \times b & = & a \times b \\ \Rightarrow & r \times a + r \times b = 0 \Rightarrow r \times (a + b) = 0 \\ \Rightarrow & r | | a + b \\ \Rightarrow & r = a + b = 3 i + j - k \end{array}$