GRAPHS

• Graph 1 — Three vectors in the \(xy\)-plane.
• Graph 2 — A vector in the \(xy\)-plane.

EXAMPLE — Vector Addition

Background & Notes

(algebraic) Two vectors with the same number of components are added component-wise:

\[\vec{a} + \vec{b} = \big< a_1 + b_1, \, a_2 + b_2, \ldots, \, a_n + b_n \big>\]

(geometric) “Follow the Path”: If the vectors being added are arranged “tail-to-head”, the vector sum is the vector whose initial point at the origin, and whose terminal point is the same as the last vector on the “path”.

Assumptions

Let \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) be three vectors in the \(xy\)-plane.

Instructions

1. Graph

Go to graph 1 and identify the vectors:

• \(\vec{u}\) (red)
• \(\vec{v}\) (green)
• \(\vec{w}\) (blue)

Copy the coordinate axes and the vectors \(\vec{u}\) and \(\vec{v}\) into your notebook.

Sketch the parallelogram spanned by \(\vec{u}\) and \(\vec{v}\), and label the four vertices of the parallelogram with their coordinates.

2. Compute

Use your diagram to find the component forms of \(\vec{u}\) and \(\vec{v}\).

Then, compute the vector sum \(\vec{u} + \vec{v}\) by adding the components of \(\vec{u}\) and \(\vec{v}\).

3. Graph & Describe

Add the vector sum \(\vec{u} + \vec{v}\) to your graph.

Describe the relationship between the vector sum \(\vec{u} + \vec{v}\) and the parallelogram spanned by \(\vec{u}\) and \(\vec{v}\).

4. Graph

Return to graph 1, and copy the coordinate axes and all three vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) into your notebook.

Use this graph to create a “path” \(\vec{u} \rightarrow \vec{v} \rightarrow \vec{w}\) where:

• \(\vec{u}\) is based (starts) at the origin.
• \(\vec{v}\) is based at the terminal point of \(\vec{u}\).
• \(\vec{w}\) is based at the terminal point of \(\vec{v}\).

On your diagram, add the vector based at the origin, whose terminal point is the end of the “path”.

This vector is the sum \(\vec{u} + \vec{v} + \vec{w}\).

5. Compute & Compare

Use your diagram to find the component form of the vector sum \(\vec{u} + \vec{v} + \vec{w}\).

Then, compute the sum directly using the component forms of \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\).

Compare your two answers — they should be the same!

6. Consider

Suppose you followed a different “path” using the same three vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\).

Do you think this would result in the same vector as \(\vec{u} + \vec{v} + \vec{w}\)? Or could it be different?

EXERCISE 1 — Scalar Multiplication

Background & Notes

In scalar multiplication, the components of a vector \(\vec{a}\) are multiplied by a scalar \(c\):

\[c \, \vec{a} = \big< c \, a_1, \, c \, a_2, \ldots, \, c \, a_n \big>\]

Assumptions

Let \(\vec{v}\) be the vector:

\[\vec{v} = \big< 1, 1 \big>\]

Instructions

1. Compute

Compute the following scalar multiples of the vector \(\vec{v} = \big< 1, 1 \big>\):

\[2 \, \vec{v}, \quad 1 \, \vec{v}, \quad \frac{1}{2} \, \vec{v}, \quad 0 \, \vec{v}, \quad -\frac{1}{2} \, \vec{v}, \quad -1 \, \vec{v}, \quad -2 \, \vec{v}\]

2. Graph

Go to graph 2, and identify the vector \(\vec{v} = \big< 1, 1 \big>\) (red).

At the top of the control window, locate the drop-down menu labeled, “Add to graph: Select…”

Add seven more vectors using the drop-down menu, by selecting “Vector: < a, b, c >” seven times.

Change the components of these vectors to those of the scalar multiples of \(c \, \vec{v}\) you just computed.

Change the \(z\)-component to zero — or delete it altogether. Change the color of a vector using the color selector.

Check the box next to each vector to activate it on the graph.

3. Compare & Conclude

On your graph, compare the directions of the vectors \(2 \, \vec{v}\), \(1 \, \vec{v}\), and \(\frac{1}{2} \, \vec{v}\) to the directions of the vectors \(-2 \, \vec{v}\), \(-1 \, \vec{v}\), and \(-\frac{1}{2} \, \vec{v}\).

What do you conclude about multiplying a (non-zero) vector by a negative scalar?

4. Compute, Compare, Conclude

Compute and compare the magnitudes of the pairs of vectors:

• \(| 2 \, \vec{v} |\) and \(| -2 \, \vec{v} |\)
• \(| 1 \, \vec{v} |\) and \(| -1 \, \vec{v} |\)
• \(| 2 \, \vec{v} |\) and \(| -2 \, \vec{v} |\)

Show that:

• \(| 2 \, \vec{v} | = | -2 \, \vec{v} | = 2 | \vec{v} |\)
• \(| 1 \, \vec{v} | = | -1 \, \vec{v} | = 1 | \vec{v} |\)
• \(| \frac{1}{2} \, \vec{v} | = | -\frac{1}{2} \, \vec{v} | = \frac{1}{2} | \vec{v} |\)

Conclude that, for a scalar \(c\) and non-zero vector \(\vec{v}\):

\[|\, c \, \vec{v} | = | c | | \vec{v} |\]

BIG IDEAS

Vector Addition (example)

Geometrically, the sum of two vectors is the diagonal of the parallelogram spanned by those vectors.

More generally, the sum of vectors can be determined by “following the path” made by basing each vector in the sum at the terminal point of the vector preceding it — the vector sum has initial point at the origin, and terminal point at the end of the “path”.

Scalar/Vector Multiplication (exercise 1)

For a non-zero vector \(\vec{a} \neq \vec{0}\), and a scalar \(c\):

Multiplying \(\vec{a}\) by \(c\) stretches or shrinks (“scales”) the magnitude of \(\vec{a}\):

\[| c \, \vec{a} | = | c || \vec{a} |\]

If \(c\) is positive, \(\vec{a}\) and \(c \, \vec{a}\) have the same direction.

If \(c\) is negative, \(\vec{a}\) and \(c \, \vec{a}\) have opposite directions.