Introduction

Extend your fluency with vectors by resolving components, combining vectors graphically and algebraically, and applying the results to real-world navigation challenges.

Problems

1. {"partTitle"=>"Part A: Component Mastery", "instructions"=>"Resolve each vector into horizontal and vertical components. Show all work and keep units consistent.", "skillCategories"=>[{"categoryTitle"=>"Component Resolution", "problems"=>[{"problemNumber"=>1, "text"=>"A displacement of 45.0 m forms a 32.0° angle above the +x-axis. Determine $d_x$ and $d_y$."}, {"problemNumber"=>2, "text"=>"An airplane experiences a velocity of 120. km/h at 18.0° south of east. Resolve the velocity into east-west and north-south components."}]}]}
2. {"partTitle"=>"Part B: Vector Synthesis", "instructions"=>"Combine multiple vectors graphically and algebraically. Compare the approaches and discuss any discrepancies.", "problems"=>[{"problemNumber"=>3, "text"=>"Two displacement vectors, $\\vec{A} = 15.0$ m at 55.0° and $\\vec{B} = 22.0$ m at 210°. Determine the magnitude and direction of $\\vec{A} + \\vec{B}$. Provide both graphical and component-based solutions."}, {"problemNumber"=>4, "text"=>"A hiker travels 2.50 km east, then 1.80 km at 40.0° north of east, and finally 0.90 km due north. What single displacement would take the hiker from start to finish?"}]}
3. {"partTitle"=>"Part C: Reflection", "instructions"=>"Discuss how vector tools support honest, precise navigation and measurement.", "problems"=>[{"problemNumber"=>5, "text"=>"How do vector components provide a universal language for describing direction and magnitude? Reference at least one of this week's labs or problems in your response."}]}