Here, we have, r1 = \(\hat i + \hat j - \hat k + \lambda(3\hat i - \hat j)\)and r2 = \(4\hat i - \hat k + \mu(2\hat i - 3\hat k)\)
To show that r1 and r2 intersects we must first equalize them
∴ r1 = r2
∴ \(\hat i + \hat j - \hat k + \lambda(3\hat i - \hat j) = 4 \hat i - \hat k + \mu(2\hat i - 3\hat k)\)
∴ \((3\lambda - 2\mu - 3)\hat i + (1 - \lambda )\hat j + (3\mu) \hat k = 0\)
Now, equating the \(\hat i, \hat j, \hat k\) each component to zero.
First equating \(\hat k\) component as zero, we get,
−3μ = 0
∴ μ = 0
Then equating \(\hat j\) component as zero, we get,
1 − λ = 0
∴ λ = 1
Lastly equating \(\hat i\) component as zero, we get,
3λ − 2μ − 3 = 0
∴ 3 − 3 = 0
Hence, r1 and r2 intersects each other.
∴ Intersecting point: μ = 0, λ = 1
Also, when we put the values of λ and μ in any of the above equation we will get the point of intersection as \(4\hat i - \hat k\).