Log-in to Explore Learning (www.explorelearning.com). Look for the Vector Gizmo. Open the Gizmo and answer the questions below. Answer these questions in a separate Word file – you do not need the questions.
Suppose you were sitting on a large, flat, thick sheet of ice, sliding around as you are pushed by a friend. Then suppose another friend joined and also started to push. Would your speed increase or decrease? Would your direction change? It depends on what direction the second friend is pushing!
A vector is a quantity with both a magnitude and a direction. Things that have both a magnitude and direction, like force, are best represented with a vector.
Describing Vectors
In this activity, you will explore vectors and ways of describing them by magnitude and direction.
1.
In the Gizmo, notice that there are two vectors on the graph, a and b . The initial point of a vector is shown by a round dot. The terminal point is shown as an arrow. You will begin this activity by exploring a . a.
Drag the initial point of a to the origin. Drag the terminal point of a to the point (0, 4). How is this vector written in the Gizmo? (Note: i is a vector that is exactly one unit long, pointing to the right, or to the "east". j is a vector that is exactly one unit long, pointing up, or to the "north".) b.
The magnitude of a vector is the distance from the initial point to the terminal point. What is the magnitude of a ? The direction of a vector is the direction in which the arrow points, from the initial point to the terminal point. Using the compass directions north, south, east, or west, what is the direction of a ? c.
If you drag the terminal point of a to form the vector 3 i + 4 j , how has the direction changed? What is the new magnitude? Click on Click to measure lengths and use the interactive ruler to check your answer. d.
Turn off the ruler. Drag a around by dragging the initial point. As you drag the vector, do the coordinates of the vector change, or do they remain 0 i + 4 j ? What do you think the coordinates of a vector represent? Does the magnitude of the vector change? Does the direction change? Explain.
2.
Drag the initial point of b to the origin and drag the terminal point of b to form the vector -3 i + 0 j . a.
Because magnitude is the size of the vector, it is always a positive quantity. What is the magnitude of b ? What is the direction of b ? b.
Drag the terminal point of b to form the vector 0 i - 5 j . Now what is the direction of b? What is the magnitude? c.
What is the direction of the vector -5 i + 5 j ? What is the magnitude? To help calculate the magnitude, sketch a right triangle.

Vector Sums
In this activity, you will explore how to find the sum of two vectors, and what it means.
1.
Suppose you are in a motorized boat traveling directly north in still waters. Set a = 0 i + 4 j to represent the velocity and direction of the boat. Suppose the boat encounters a strong current in the water, flowing from west to east. Set b = 3 i + 0 j to represent the velocity and direction of the current. a.
Where do you think your boat will go? Make a sketch to show your guess. b.
The resulting speed and direction of the boat is represented by the sum, or resultant, of a and b.
Click on Show resultant. The resultant is represented by c on the graph. What are the coordinates of c? c.
Increase the force of the current by dragging b to 6 i + 0 j . What is the effect on c ? d.
What if the current were suddenly directly against you? Set b = 0 i - 2 j to simulate this. What is c , the resultant or sum of a and b now? What does this tell you about the speed and direction of your boat? e.
Make a conjecture about how the coordinates of c are obtained from the coordinates of a and b .
Click on Show sum computation to check your hypothesis. f.
Vary the coordinates of a and b to convince yourself that adding vectors simply means adding the coordinates of the vectors. Notice that changing the location of a or b does not change what the resultant is.
2.
Turn off Show resultant. Place the initial point of a at the origin and set a = -2 i + 3 j . Place the initial point of b on the terminal point of a and set b = -4 i + 1 j . a.
Calculate a + b by adding the coordinates of a and b . How does that answer compare to the coordinates of the terminal point of b ? Click on Show resultant and Show sum computation to check your answers. b.
Turn on Show x, y components. How does the sum of the x components of a and b relate to the x component of a + b ? How about the y components? Try other vectors. Is this always true? c.
You should have found that a + b = -6 i + 4 j . If you were to draw this vector on the graph, with its initial point at the origin, where would its terminal point lie? How does that compare with where the terminal point of b lies? d.
If you move vectors a and b around on the graph, but leave them the same vector (move them by their initial point), does that change the value of the resultant (sum)? Explain.

3.
Use the Gizmo and what you have learned about vector addition and dot products to find the following answers. a.
Find 1 i + 4 j + (-5 i - 3 j ). b.
What is the resultant of a = 0 i + 4 j and b = -3 i + 0 j ? c.
Find the sum of a = -5 i + 0 j and b = -2 i + 0 j . d.
What is c if c = a + b and a = 2 i + 3 j and b = -2 i - 3 j ? This is called a state of equilibrium. e.
Name another pair of vectors that create a state of equilibrium.