Math9999 said:Homework Statement
Find the vector form of the equation of the line in ℝ2 that passes through P=(2, -1) and is parallel to the line with general equation 2x-3y=1.
Homework Equations
None.
The Attempt at a Solution
vector form: x=p+td
general form: ax+by=c
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a=2, b=-3, so [x, y]=[2, -1]+t[2, -3], am I right?
Math9999 said:They are but how do I figure out the normal vector?
You can (and should) check this yourself. Clearly, the line from your equation goes through the point (2, -1). Do you see why? But does it have the right slope?Math9999 said:a=2, b=-3, so [x, y]=[2, -1]+t[2, -3], am I right?
Why do you need the normal vector?Math9999 said:So how do I figure out the normal vector?
Yes, that's the right slope. Now you should be able to get the equation in vector form.Math9999 said:So 2x-3y=1, 3y=2x-1, y=2/3x-1/3 and m=2/3, which is the slope?
What's the slope of the vector <3, 2>? Try drawing it in the plane.Math9999 said:But how? I still don't get it. If the right slope is m=2/3, then how is the answer [x, y]=[2, -1]+t[3, 2]?
YesMath9999 said:m=rise/run=2/3?
Post #9Math9999 said:Find the vector form of the equation of the line in ℝ2 that passes through P=(2, -1) and is parallel to the line with general equation 2x-3y=1.
Is there some value of t for which <x, y> = <2, -1>? IOW, does the line pass through (2, -1)?Math9999 said:If the right slope is m=2/3, then how is the answer [x, y]=[2, -1]+t[3, 2]?
A vector form is a way of representing a line or a plane in a coordinate system using a vector equation. It is often used in mathematics and physics to describe the motion of objects in a given space.
To find the vector form of a line, you need to know a point on the line and a vector that is parallel to the line. The vector form is written as r = a + tb, where r is a position vector, a is a point on the line, and b is the direction vector.
Two lines are parallel if they have the same slope and will never intersect. This means that they will have the same direction vector and will be a certain distance apart from each other.
A line is parallel to 2x-3y=1 if its slope is equal to the slope of the given line. To find the slope of a line, we use the formula y = mx + b, where m is the slope. In this case, the slope is 2/3. Therefore, any line with a slope of 2/3 will be parallel to 2x-3y=1.
The equation of the line passing through (2, -1) and parallel to 2x-3y=1 can be found using the point-slope form. We know the slope is 2/3 and the point (2, -1) is on the line. Therefore, the equation is y - (-1) = 2/3(x - 2), which simplifies to y = 2/3x - 5/3.