there is a straight line 2x+y=10 on the Cartesian plane. Can you explain clearly how do I turn it into the vector form of [1 -2]?
To turn the equation of a straight line in the Cartesian plane into vector form, we need to first rewrite the equation in slope-intercept form, which is in the form y = mx + b. Here's how:
2x + y = 10
Rearranging this equation by isolating y, we get:
y = -2x + 10
Now, we can see that the slope of the line is -2, and the y-intercept is 10.
To convert this into vector form, we need to find two points on the line. Let's choose x = 0 and x = 1:
For x = 0:
y = -2(0) + 10
y = 10
So, one point on the line is (0, 10).
For x = 1:
y = -2(1) + 10
y = 8
Thus, another point on the line is (1, 8).
Next, we can compute the difference between these two points to find the directional vector:
[1, 8] - [0, 10] = [1-0, 8-10] = [1, -2]
Thus, the vector form of the line is [1, -2].