GRAPHS

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• Graph 1 — Two vectors, the plane they span, and their cross product.
• Graph 2 — Two vectors in the \(xy\)-plane, and their cross product. There are sliders to change magnitudes and the angle between the vectors.

EXERCISE 1 — Reversing the Order of Vectors in a Cross Product

Background & Notes

The cross product is a vector operation used exclusively for vectors in 3-space:

\[\begin{align*}
\vec{v} &= \big< v_1, v_2, v_3 \big> = v_1 \hat{\imath} + v_2 \hat{\jmath} + v_3 \hat{k}\\
\vec{w} &= \big< w_1, w_2, w_3 \big> = w_1 \hat{\imath} + w_2 \hat{\jmath} + w_3 \hat{k}
\end{align*}\]

\[\vec{v} \times \vec{w} = (v_2 w_3 - v_3 w_2) \hat{\imath} - (v_1 w_3 - v_3 w_1) \hat{\jmath} + (v_1 w_2 - v_2 w_1) \hat{k}\]

There are several valid methods of computing the cross product. One of these is the determinant method:

\[\vec{v} \times \vec{w} = \left| \begin{array}{c c c} \hat{\imath} & \hat{\jmath} & \hat{k}\\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{array} \right|\]

Since \(\vec{v} \times \vec{w}\) is a vector, it has both magnitude and direction.

This exercise explores what happens to components, magnitude, and direction when you reverse the order of the vectors in a cross product.

Assumptions

Let \(\vec{v}\) and \(\vec{w}\) be the 3-dimensional vectors:

\[\begin{align*}
\vec{v} = \big< 1, 0, -1 \big> &= \hat{\imath} - \hat{k}\\
\vec{w} = \big< 0, 1, -1 \big> &= \hat{\jmath} - \hat{k}
\end{align*}\]

Instructions

1. Compute \(\vec{v} \times \vec{w}\) (example)

Compute \(\vec{v} \times \vec{w}\) using the determinant method:

\[\vec{v} \times \vec{w} = \left| \begin{array}{c c c} \hat{\imath} & \hat{\jmath} & \hat{k}\\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array} \right|\]

2. Graph \(\vec{v} \times \vec{w}\) (example)

Go to graph 1 and identify the vectors:

• \(\vec{v} = \hat{\imath} - \hat{k}\) (red)
• \(\vec{w} = \hat{\jmath} - \hat{k}\) (blue)
• \(\vec{v} \times \vec{w} = \hat{\imath} + \hat{\jmath} + \hat{k}\) (bright green).

Rotate the graph as needed to get a good view.

3. Compute & Graph \(\vec{w} \times \vec{v}\)

Compute \(\vec{w} \times \vec{v}\) using the determinant method:

\[\vec{w} \times \vec{v} = \left| \begin{array}{c c } \hat{\imath} & \hat{\jmath} & \hat{k}\\ 0 & 1 & -1 \\ 1 & 0 & -1 \end{array} \right|\]

Return to graph 1.

On the graph, identify the vectors \(\vec{v}\) (red), \(\vec{w}\) (blue), and \(\vec{v} \times \vec{w}\) (bright green).

In the command window, locate the text entry box for the vector < 0, 0, 0 > (dark green), and replace its components with those the cross product \(\vec{w} \times \vec{v}\).

4. Compare & Summarize

Compare Component Forms — Compare the components of the vectors \(\vec{w} \times \vec{v}\) and \(\vec{v} \times \vec{w}\).

Compare Magnitudes — Compute and compare the magnitudes \(| \vec{v} \times \vec{w} |\) and \(| \vec{w} \times \vec{v}|\).

Compare Directions — In graph 1, compare the directions of the vectors \(\vec{v} \times \vec{w}\) and \(\vec{w} \times \vec{v}\).

Summarize Your Findings: Comparing \(\vec{v} \times \vec{w}\) and \(\vec{w} \times \vec{v}\):

• How are they related algebraically?
• How are their magnitudes related?
• How are their directions related?

EXERCISE 2 — The Cross Product & Scalar Multiplication

Background & Notes

This exercise explores the interaction of scalar multiplication and the cross product.

Assumptions

Let \(\vec{v}\), \(\vec{u}\), and \(\vec{w}\) be the vectors:

\[\begin{align*}
\vec{u} = \big< 5, 0, 7\big> &= 5 \hat{\imath} + 7 \hat {k}\\
\vec{v} = \big< 1, 2, 3 \big> &= \hat{\imath} + 2 \hat{\jmath} + 3 \hat{k}\\
\vec{w} = \big< -2, -4, -6 \big> &= -2 \hat{\imath} - 4 \hat{\jmath} - 6 \hat {k}
\end{align*}\]

Instructions

1. Graph (example)

Go to graph 2, and identify the vectors \(\vec{a}\) (red), \(\vec{b}\) (blue) and the cross product \(\vec{a} \times \vec{b}\) (green).

Locate the slider labeled c at the top of the control window.

The magnitude of \(\vec{a}\) (red) is fixed (stays the same).

The c-slider controls the magnitude of \(\vec{b}\) (blue).

Observe how changes to the magnitude of \(\vec{b}\) affect the magnitude of the cross product \(\vec{a} \times \vec{b}\) (green).

Rotate the graph as needed to get a good view!

2. Identify (example)

Observe that \(\vec{w} = -2 \hat{\imath} - 4 \hat{\jmath} - 6 \hat{k}\) is a scalar multiple of \(\vec{v} = \hat{\imath} + 2 \hat{\jmath} + 3 \hat{k}\):

\[\vec{w} = c \vec{v}\]

What is the value of the scalar \(c\)?

3. Compute (example)

Compute the cross products \(\vec{u} \times \vec{v}\) and \(\vec{u} \times \vec{w}\), using the determinant method.

4. Compare & Analyze

Compare the components of \(\vec{u} \times \vec{v}\) and \(\vec{u} \times \vec{w}\).

Keep in mind: \(\vec{w} = -2 \vec{v}\), so:

\[\vec{u} \times \vec{w} = \vec{u} \times (-2 \vec{v})\]

Question: What role is played by the scalar \(c = -2\)?

5. Challenge — Formalize

Working symbolically, let \(c\) be a scalar, and let:

\[\begin{align*}
\vec{a} = \big< a_1, a_2, a_3 \big> &= a_1 \hat{\imath} + a_2 \hat{\jmath} + a_3 \hat{k}\\
\vec{b} = \big< b_1, b_2, b_3 \big> &= b_1 \hat{\imath} + b_2 \hat{\jmath} + b_3 \hat{k}
\end{align*}\]

Using the determinant method, compute:

\[\big( c \vec{a} \big) \times \vec{b}\]

and:

\[c \big( \vec{a} \times \vec{b} \big)\]

to show they are equal:

\[\big( c \vec{a} \big) \times \vec{b} = c \big( \vec{a} \times \vec{b} \big)\]

Conclude that scalar multipliers can be factored out of the cross product.

EXERCISE 3 — The Cross Product & Parallel Vectors

Background & Notes

Recall that two (non-zero) vectors are parallel if and only if they are scalar multiples of each other:

\[\vec{a} \| \vec{b} \quad \iff \quad \vec{b} = c \vec{a}\]

This exercise explores what happens when you compute the cross product of parallel vectors.

Assumptions

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

\[\vec{v} = \big< 2, 1, -5 \big> = 2 \hat{\imath} + \hat{\jmath} - 5 \hat{k}\]

Instructions

1. Graph

Return to graph 2, and identify the vectors \(\vec{a}\) (red), \(\vec{b}\) (blue) and the cross product \(\vec{a} \times \vec{b}\) (green).

Locate the slider labeled a at the top of the control window.

Use the a-slider to change the angle between the vectors \(\vec{a}\) and \(\vec{b}\), and observe what happens to the cross product \(\vec{a} \times \vec{b}\) when \(\vec{a}\) and \(\vec{b}\) are parallel.

Rotate the graph as needed to get a good view!

2. Construct Example

Choose a non-zero scalar \(c \neq 0\), and construct the vector \(\vec{w} = c \vec{v}\).

Observe: Since \(\vec{w}\) is a scalar multiple of \(\vec{v}\), the vectors \(\vec{w}\) and \(\vec{v}\) are parallel.

3. Compute

Using the determinant method, compute \(\vec{w} \times \vec{v}\).

Keep in mind: \(\vec{w} \| \vec{v}\)

4. Hypothesize

Based on your observations, what can you speculate about the cross product of parallel vectors?

5. Challenge — Formalize

Working symbolically, let \(c \neq 0\) be a non-zero scalar, and \(\vec{v} \neq \vec{0}\) a non-zero vector:

\[\vec{v} = \big< v_1, v_2, v_3 \big> = v_1 \hat{\imath} + v_2 \hat{\jmath} + v_3 \hat{k}\]

Using the determinant method, compute \(\big( c \vec{v} \big) \times \vec{v}\) to show that the cross product of parallel vectors is the zero vector:

\[\big( c \vec{v} \big) \times \vec{v} = \vec{0}\]

BIG IDEAS

Reversing the Order of Vectors in a Cross Product (exercise 1)

The cross products \(\vec{a} \times \vec{b}\) and \(\vec{b} \times \vec{a}\):

• Algebraically — Are related by multiplication by -1:

\[\vec{b} \times \vec{a} = - ( \vec{a} \times \vec{b} )\]

• Geometrically — Have the same magnitude, but opposite directions.

The Cross Product & Scalar Multiplication (exercise 2)

Scalar multiplication factors through a cross product:

\[\big( c \vec{a} \big) \times \vec{b} = \vec{a} \times \big( c \vec{b} \big) = c \big( \vec{a} \times \vec{b} \big)\]

In other words:

You can multiply \(\vec{a}\) by \(c\), then compute the cross product:

\[(c \vec{a}) \times \vec{b}\]

or: You can multiply \(\vec{b}\) by \(c\), then compute the cross product:

\[\vec{a} \times (c \vec{b})\]

or: You can compute the cross product \(\vec{a} \times \vec{b}\), then multiply by $c$:

\[c (\vec{a} \times \vec{b})\]

The Cross Product & Parallel Vectors (exercise 3)

If \(\vec{a}\) and \(\vec{b}\) are parallel, then:

\[\vec{a} \times \vec{b} = \vec{0} = \big< 0, 0, 0 \big>\]