Angle Between Two Lines

Step 1: Understanding the Concept:
The angle between two lines with direction vectors and satisfies .

Step 2: Detailed Explanation:
, .
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Step 3: Final Answer:
The angle is .

Step 1: Plot the constraints.
We are given the following constraints: 1. 2. 3. These represent linear inequalities. We first solve for the boundaries of the inequalities.
Step 2: Find the points of intersection.
From the first constraint , we get the line . From the second constraint , we get the line . To find the points of intersection, solve the system of equations: Subtract the first equation from the second: Substitute into : Thus, the intersection point is .
Step 3: Evaluate at the vertices.
The feasible region is formed by the constraints. The vertices are the points of intersection of the lines and the axes. We now evaluate at the points , , and . - At : - At : - At :
Step 4: Conclusion.
The maximum value of is 40 at .