Write as a Vector Equality y=2x+9 , y=2x-9

Problem

y=2*x+9,y=2*x−9

Solution

1. Identify the general vector form of a line in ℝ2 which is r=(r_0)+t*v where (r_0) is a position vector to a point on the line and v is the direction vector.

2. Determine the direction vector for both lines. Since both lines have a slope of m=2 a change of 1 in x results in a change of 2 in y Thus, the direction vector for both is v=([1],[2])

3. Find a point on the first line y=2*x+9 Setting x=0 gives y=9 so a position vector is (r_1)=([0],[9])

4. Find a point on the second line y=2*x−9 Setting x=0 gives y=−9 so a position vector is (r_2)=([0],[−9])

5. Express the lines as vector equations using a parameter t

Final Answer

([x],[y])=([0],[9])+t*([1],[2]),([x],[y])=([0],[−9])+t*([1],[2])

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