Lesson Plan - Vector Addition

Suggestions for Instructors

The purpose of this lesson is to cover the addition of vectors. Both graphical methods and component methods are covered. For students who have not studied vectors before, you might wish to start with the vector lesson.

One of the most common problem solving mistake students will make in physics is the failure to put a problem into vector form. They often will assume that they can just add magnitudes, even if they are not in the same direction. Try to have students draw each problem they work on before they try to solve it, including drawing arrows for any vector quantities to show magnitude and direction.

Standards

NCTM Grades 6-8 Geometry:

• Specify locations and describe spatial relationships using coordinate geometry and other representational systems
  • Use coordinate geometry to represent and examine the properties of geometric shapes;
  • Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
• Use visualization, spatial reasoning, and geometric modeling to solve problems
  • Draw geometric objects with specified properties, such as side lengths or angle measures;
  • Use visual tools such as networks to represent and solve problems;
  • Use geometric models to represent and explain numerical and algebraic relationships;
  • Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Solutions

1. 10 meters per second, at 0.64 radians (37 degrees)
2. 95 kph, at 0.148 radians west of north (8 degrees west of north)
3. In many ways, most problems are poorly solved by graphical methods because of the uncertainty in drawing and measuring lines, however, problems which lend themselves naturally to being expressed in vector components are much easier to solve when solved for each component separately. However, even for these problems, the graphical solution can help to understand how to set up a solution using components.
4. The actual speed of the boat as observed from someone on the shore (AS) is equal to the speed of the boat with respect to the water (SB) plus the speed of the water (SW), added as vectors. If the shore lies in the x direction, then the x component of the actual speed (AS) should be zero, as the boat needs to go straight across the river. The magnitude of the sped of the boat with respect to the water is 20. So, the x component of the speed of the boat should be equal in magnitude and opposite in direction to the speed of the water.
   • (state equation) SB_x = - SW_x
   • (compute each component) 20 cos (theta) = 5 (water is flowing downstream, SW_x = -5)
   • (solve equation) theta = acos(5/20)
   • theta = acos(5/20)
   • theta = 75 degrees from shore
5. Upstream, opposite the water
6. Depending on how you count a "step", the above method is three steps. If you count a step as each equation to be solved, this is done with a single equation. If you count each mathematical operation, the number is a little higher. Your students mileage will vary.