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# C4 - Vectors for parallel lines

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1. Just a little thing I don't understand. I'm using this video to revise vectors.

So when you have two parallel lines, and you have variables a and b in their direction vectors, you can sometimes get 2 different values of a and b. Why is this? Are the two sets of variables for two different parallel lines? Or can two different values for a and b be subbed in to give the same line?

It's hard to explain what I mean, but if you watch the example question in the video you should see it.
2. (Original post by JordanL_)
Just a little thing I don't understand. I'm using this video to revise vectors.

So when you have two parallel lines, and you have variables a and b in their direction vectors, you can sometimes get 2 different values of a and b. Why is this? Are the two sets of variables for two different parallel lines? Or can two different values for a and b be subbed in to give the same line?

It's hard to explain what I mean, but if you watch the example question in the video you should see it.
If the direction vectors of two lines are identical, or multiples of one another, the lines are parallel
3. (Original post by JordanL_)
Just a little thing I don't understand. I'm using this video to revise vectors.

So when you have two parallel lines, and you have variables a and b in their direction vectors, you can sometimes get 2 different values of a and b. Why is this? Are the two sets of variables for two different parallel lines? Or can two different values for a and b be subbed in to give the same line?

It's hard to explain what I mean, but if you watch the example question in the video you should see it.
If they are two parallel lines, that means their directions are multiplies of one another.

So for example, the vector (1, 2) is parallel to the vector (2, 4) = 2(1, 2). You can plot those two vectors yourself to see they're parallel.

In the same vein, for two lines to be parallel, their direction vectors must be parallel, so if you have r = a + bc and r = d + ef.

Then the point they start from are utterly irrelevant, you don't care about a or d at all.

What you do care about is that their directions are roughly the same, that is, they should point in the same direction. Which is just saying that the vector b and the vector f, should be multiples of one another, not just equal to one another.

So b = 2f works, b = 1.5f also works, etc... (so does b = f).
4. (Original post by Zacken)
If they are two parallel lines, that means their directions are multiplies of one another.

So for example, the vector (1, 2) is parallel to the vector (2, 4) = 2(1, 2). You can plot those two vectors yourself to see they're parallel.

In the same vein, for two lines to be parallel, their direction vectors must be parallel, so if you have r = a + bc and r = d + ef.

Then the point they start from are utterly irrelevant, you don't care about a or d at all.

What you do care about is that their directions are roughly the same, that is, they should point in the same direction. Which is just saying that the vector b and the vector f, should be multiples of one another, not just equal to one another.

So b = 2f works, b = 1.5f also works, etc... (so does b = f).
Thank you! But what if you have

r = a + v(c - 1, -c - 1, d)
r = b + w(2c, 3 - 5c, 15)

You solve to find the values of c and d. But you get two values of c due to a quadratic, and then two values of d because you have two values of c. What does this mean?
5. Having watched the video, a,b are variables in the coordinates of the direction vectors of two lines, which you're told are parallel.

In this case, the two sets of values of a,b give two different sets of parallel lines.

Note that a,b are nothing to do with the scalar multiplier of the direction vector, as such.

In essence each line here is defined by a fixed point, and a variable direction vector. It just so happens in this question that there are two possibilities that will allow the direction vectors to line up, and give parallel lines.

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