00:01
In this question, we are asked to determine if the given lines are parallel, perpendicular or neither, and if they are not parallel, we are asked to find the point of intersection.
00:10
And the first step would be to find the equations of the lines.
00:13
The first line is the line through the origin and the point 2 -0 -negative 1.
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So let's call the origin.
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Let's give it a name.
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Let's call it o with coordinate 0 -0.
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And let's call the second point, let's say, p.
00:36
Also, let's give names for points on the line l2.
00:40
Let's call them q and r.
00:46
Now, let's get back to the first line.
00:51
The equation of the first line, let's call it r1 of c, is x, y, and z, knot, plus t, multiplied by v1, where v1 is a direction vector of x.
01:08
Of the first line.
01:10
X, y, knot and z not could be coordinates of any point on the line.
01:15
For example, we can choose the origin.
01:19
For x, y, not and z not, and then we can rewrite the equation of the line as r1 of t equals to triple zero plus t times v1, which simplifies to t times v1.
01:34
Now to find the direction vector, for the direction vector, it could be any vector parallel to our line.
01:43
And we can take the vector connecting the points o and p.
01:50
To get the coordinates of the vector o we need to subtract the coordinates of o from the coordinates of p.
01:57
And we are just going to get 2 -0 negative 1.
02:03
Therefore the equation of the line or the first line is multiplied by 2, 0 and negative 1.
02:14
Next let's find the equation of the second line.
02:19
Let's call it r2 of s.
02:24
A point lying on the second line let's choose the point q for example 1 negative 1 1 plus t multiply sorry plus s multiplied by v2 where v2 is a direction vector of the second line and we can take for v2 the vector connecting the points q and r sorry there should be no arrows above the points to get the coordinates of the vector q r we need to subtract the coordinates of q from the coordinates of r so we we need to subtract 1 from 4, then the second coordinate is going to be 1 minus negative 1, and the 3 minus 1.
03:28
This equals to 3, 2 and 2.
03:33
These are the coordinates of the vector qr.
03:39
Now two lines are parallel.
03:42
So now let's write down the equation of the second line.
03:46
Are 2 of s equals to 1 negative 1, 1 plus s times 2.
03:55
3, 2, all right.
04:02
Now, two vectors, two lines are parallel, if their direction vectors are parallel.
04:19
If v1 and v2 are parallel.
04:23
Now, v1 and v2 are parallel if there are multiples of each other.
04:35
So, we 1 and v2 are parallel if v1 equals to some constant k multiplied by v2.
04:53
Or in our case, we want, recall that v1 equals to 2, 0 ,000.
04:59
Negative 1 and we want this to be equal to k multiplied by v2 with coordinates 322...