How would I find the direction vector parallel to a line? For example, how is (2,1) a direction vector parallel to the line -x+2y-10=0
To find a direction vector parallel to a line, you need to manipulate the equation of the line into slope-intercept form (y = mx + b) or convert it into a parametric form (x = x1 + ma, y = y1 + mb).
Let's go through the process using your example:
1. Rewrite the equation -x + 2y - 10 = 0 in slope-intercept form, y = mx + b:
Add x to both sides: 2y = x + 10
Divide everything by 2: y = (1/2)x + 5/2
2. Now we have the equation in slope-intercept form, where the coefficient of x, in this case, 1/2, is the slope of the line. A direction vector parallel to the line can be created using this slope.
3. Create a vector using the slope (1/2):
For a direction vector, choose any non-zero multiple of the slope. A common choice is to use: (2, 1), (-2, -1), (4, 2), etc.
Therefore, (2, 1) is one possible direction vector parallel to the line -x + 2y - 10 = 0.