Write as a Vector Equality

Problem

y=2*x+3,y=−1/2*x+5.5,y=2*x+4,y=−1/2*x+3.5

Solution

1. Identify the slope-intercept form y=m*x+b for each line to determine the direction vectors.

2. Determine the direction vector v for each line. If the slope is m the direction vector can be written as ([1],[m])

3. Identify a position vector (r_0) for each line using the y-intercept (0,b) which gives (r_0)=([0],[b])

4. Construct the vector equation r=(r_0)+t*v for each given line.

5. Convert the decimal 5.5 to the fraction 11/2 and 3.5 to 7/2 for consistency.

6. Simplify the direction vectors by multiplying by a scalar to remove fractions where applicable (e.g., ([1],[−1/2]) becomes ([2],[−1]).

Final Answer

([x],[y])=([0],[3])+t*([1],[2])

([x],[y])=([0],[11/2])+s*([2],[−1])

([x],[y])=([0],[4])+u*([1],[2])

([x],[y])=([0],[7/2])+v*([2],[−1])

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