Do the following vector equations intersect? l = (1 + μ)i + (2 - μ)j + (2μ - 5)k, and m = 2λi + 3j + (2 + λ)k.

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Firstly we must appreciate that if there is an intersection between l and m, this means that there exists a point that occurs on both lines. Therefore if there is an intersection, it is when l = m.

So we must start by making the two vector equations equal to eachother:

(1 + μ)i + (2 - μ)j +(2μ - 5)k = 2λi + 3j + (2 + λ)k

From here we can equate the coefficients to make 3 simultaneous equations with the variables λ and μ. This means making the coefficients of i equal to eachother, the coefficients of j equal to eachother and the coefficients of k equal to eachother. This gives us the following equations:

1 + μ = 2λ (equation 1),

2 - μ = 3 (equation 2),

2μ - 5 = 2 + λ (equation 3).

From equation 2 we can see that μ = -1. Using this value of μ, we can solve for λ in equation 1 using substitution, which gives:

1 - 1 = 2λ, so 2λ = 0, giving λ = 0.

So now we have a value for λ and a value for μ. Here we can show whether the two vectors intersect or not by substituting these values into equation 3. The two vectors intersect if and only if doing this gives a true statement (e.g 1 + 1 = 2).

So by substituting μ = -1 and λ = 0 into the equation 2μ - 5 = 2 + λ, we get:

2(-1) - 5 = 2 + 0, which is the same as

-7 = 2.

Obviously, this is not a true statement and therefore we can conclude that the two lines do not intersect at any point.

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