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# C4 Vectors question watch

1. The line l-one has the vector equation
r=11i+5j+6k+ T(4i+2j+4k)
The line l-two has the vector equation
r=24i+4j+13k+S(7i+j+5k)

Given that theta is the acute angle between l-one and l-two, find the value of costheta

So I know Im supposed to use costheta=a.b/(|a||b|)

But why is it that we use the direction vector part of both lines? Can someone explain in detail?
2. well the angle between two lines doesnt depend on where they are in space, it depends on where each line is pointing. this is what the direction vector tells you..
3. (Original post by RK92)
well the angle between two lines doesnt depend on where they are in space, it depends on where each line is pointing. this is what the direction vector tells you..
Im finding it difficult to visualise the direction vectors of the line equations meeting ...and what about the scaling factor...?
Should i just assume they meet since the question says so?
4. It is possible for the two lines to be skew, not to interesct, if you want to prove they intersect you could solve them as 3 simultaneous equations. Or its probably easier to assume they intersect given that the question requires them to. Just solve using the formula that you gave...
5. (Original post by confuzzled92)
Im finding it difficult to visualise the direction vectors of the line equations meeting ...and what about the scaling factor...?
Should i just assume they meet since the question says so?
This is hard to explain without a moving diagram/model but essentially, direction vectors are free to move around in vector space. They don't need to meet to have an angle between them, but they do need to be non-parallel.

Do you have any straws or sticks etc. at home? If so, grab a pair of them and leave one on a surface and hold the other one in the air so they are not touching and they are non-parallel. Then look down on them from above. Notice that, no matter how much you move them (without rotating them), the angle between them remains the same.

The scaling factor is used to map to a particular point on the line but it doesn't affect the angle between them. If you don't believe me, get another straw and add it to the end of one of the other two (parallel to it) that you already had. You'll see that the angle stays the same.
6. thanks! That was a great explanation.

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