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

10 meters per second, at 0.64 radians (37 degrees)

95 kph, at 0.148 radians west of north (8 degrees west of north)

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.

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

Upstream, opposite the water

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.