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+2 votes
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in Mathematics by (31.5k points)
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Show that the lines vector r = i + j - k + λ(3i - j) and vector r = 4i - k + μ(2i +3k) intersect. Also find point of intersection.

2 Answers

+1 vote
by (15.0k points)
selected by
 
Best answer

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 \(4\hat i - \hat k\).

+2 votes
by (61.2k points)

General points on the lines are

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