Student Exploration: Vectors
Vocabulary: component, dot product, magnitude, resultant, scalar, unit vector notation, vector
Prior Knowledge Question (Do this BEFORE using the Gizmo.)
An airplane is traveling north at 300 km/h. Suddenly, it is hit by a strong crosswind blowing 150 km/h from west to east.
Draw an arrow on the diagram showing the direction you think the plane will most likely move. Explain your answer.
Displacement, velocity, momentum, acceleration, and force are all examples of quantities that have both direction and magnitude. Anything with direction and magnitude can be represented using a vector.
Look at vectors a and b on the Vectors Gizmo™ grid. The initial point of each vector is shown with a circle. The terminal point of each vector is located at the tip of the arrow. Each vector is described by two components: the i component and the jcomponent.
The two components written together make up the unit vector notation. What is the unit vector notation of vector a?
Move the initial point of vector a to the origin (0, 0) on the grid
How did the components of vector a change?
Drag the terminal point of vector a so that it lines up with the x-axis. Which component describes the vector’s position along the x-axis? B=3i-j
Drag the terminal point of a so that it lines up with the y-axis. Which component describes the vector’s position along the y-axis? Still the same
Vector magnitude and angle
Get the Gizmo ready:
Change vector a so that its notation is 0i + 3j.
You will need a scientific calculator for this activity.
Question: How can you determine a vector’s magnitude and angle?
Observe: The magnitude of a vector is the distance from the vector’s initial point to its terminal point. The magnitude of a vector is written: ||x||. Magnitude is a scalar, or a number that does not indicate direction.
What is the magnitude of vector a?
Turn on Click to measure lengths and use the ruler to check your answer.
Turn off the Ruler. Drag the tip of vector a so that its notation is 4i + 3j. What do you think the magnitude of vector a is now?
Explore: A vector can be broken down into perpendicular vectors that describe its length along the x and y axes. Turn on Show x, y components. How do the x and y vectors that appearfor vector a relate to the i and j notation?
Calculate: The x, y components of vector a form the two sides of a right triangle. The length of the hypotenuse of that triangle will equal the length (and, thus, the magnitude) of vector a.
Use the Pythagorean theorem to calculate the magnitude of vector a.