Along the line 3x+4y=-2 in the direction of decreasing y what is the vector of this line ? I don't understand how they say the vector is (4,-3)
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So for every units you have gone to the right, you have gone units down.
This would suggest that this line would have direction the same as .
But we don't know the length of this vector. So multiply the vector by some parameter .
Therefore the vector that goes in the direction of this line that also has the same length as this line can be written as
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Now we just need a point that lies on the line so we know where to place this vector. The intercept will suffice.
Therefore, the vector form of this line is
or if you like
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Here's another way to do this:
The equation of the line is 3x+4y= 2. If x= 0, 4y= 2 so y= 1/2. One point on the line is (0, 1/2). If y= 0, 3x= 2 so x= 2/3. Another point on that line is (2/3, 0).
Since 0< 1/2, going in the direction of decreasing y means going from (0, 1/2) to (2/3, 0). A vector from (0,1/2) to (2/3, 0) is .
Since multiplying a vector by any number gives another vector in that direction, we can get rid of the fractions by multiplying the vector by 6: .