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)
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
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
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: .