Write as a Vector Equality y=x+3 , y=-1/2x-3

Problem

{[y=x+3],[y=−1/2*x−3])

Solution

1. Identify the standard form of a vector equation for a line, which is r=a+t*v where a is a position vector of a point on the line and v is the direction vector.

2. Determine a point and direction for the first line y=x+3 When x=0 y=3 giving point (0,3) The slope m=1 implies that for every 1 unit increase in x y increases by 1 giving direction vector ([1],[1])

3. Determine a point and direction for the second line y=−1/2*x−3 When x=0 y=−3 giving point (0,−3) The slope m=−1/2 implies that for every 2 units increase in x y decreases by 1 giving direction vector ([2],[−1])

4. Equate the two vector expressions to represent the system as a vector equality.

Final Answer

([0],[3])+t*([1],[1])=([0],[−3])+s*([2],[−1])

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