GRAPHS & CODE

Graphs

To reset a graph, refresh its browser window.

• Graph 1 — Geometric properties of the cross product: normal vector and area.
• Graph 2 — The magnitude of the cross product \(\vec{v} \times \vec{w}\), and the area of the parallelogram spanned by \(\vec{v}\) and \(\vec{w}\).
• Graph 3 — The direction of \(\vec{v} \times \vec{w}\) relative to \(\vec{v}\) and \(\vec{w}\) (orientation) — with \(\vec{v}\) and \(\vec{w}\) in the \(xy\)-plane.
• Graph 4 — Four vectors in 3-space.

Code

Use this code to compute the cross product \(\vec{v} \times \vec{w}\):

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

and its magnitude:

\[| \vec{v} \times \vec{w} | = |\vec{v}| |\vec{w}| \sin \theta\]

EXERCISE 1 — The Magnitude of the Cross Product & Area

Background & Notes

A key geometric property of the cross product is that its magnitude gives us a way to measure area.

If \(\vec{a}, \vec{b} \neq \vec{0}\) are vectors in 3-space that are not parallel, the magnitude of their cross product is the area of the parallelogram spanned by \(\vec{a}\) and \(\vec{b}\):

\[|\vec{a} \times \vec{b}| = | \vec{a} | | \vec{b} | \sin \theta\]

In this exercise:

• \(\vec{v}\) and \(\vec{w}\) span the plane \(x + y + z = 0\).
• The parallelogram spanned by \(\vec{v}\) and \(\vec{w}\) lies in the plane \(x + y + z = 0\).
• The magnitude \(|\vec{v} \times \vec{w}|\) equals the area of that parallelogram.

Assumptions

Let \(\vec{v}\) and \(\vec{w}\) be the 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. Graph (example)

Go to graph 1, and identify:

• The vectors \(\vec{v}\) (red) and \(\vec{w}\) (blue).
• The cross product \(\vec{v} \times \vec{w}\) (green).
• The plane \(x + y + z = 0\) spanned by \(\vec{v}\) and \(\vec{w}\).
• The parallelogram spanned by \(\vec{v}\) and \(\vec{w}\), in the plane \(x + y + z = 0\).

2. Compute — Area of Parallelogram = Magnitude of Cross Product (example)

The geometric definition of \(\vec{v} \times \vec{w}\) is:

\[|\vec{v} \times \vec{w}| = |\vec{v}| |\vec{w}| \sin\theta\]

From trigonometry, the area of the parallelogram with sides \(\vec{v}\) and \(\vec{w}\), is:

\[A = |\vec{v}| |\vec{w}| \sin\theta\]

This means you can find the area of the parallelogram in graph 1 by computing \(|\vec{v} \times \vec{w}|\).

Edit and run this code to compute the magnitude of the cross product \(|\vec{v} \times \vec{w}|\):

• Define the vectors v = sp.Matrix([a,b,c]) and w = sp.Matrix([a,b,c]) using the components of \(\vec{v}\) and \(\vec{w}\).
• Run the code, and make note of the resulting magnitude \(|\vec{v} \times \vec{w}|\).

3. Graph, Estimate, Confirm

Go to graph 2.

In the graph, identify:

• The vectors \(\vec{a}\) (red) and \(\vec{b}\) (blue).
• The cross product \(\vec{a} \times \vec{b}\) (green).
• The parallelogram spanned by \(\vec{a}\) and \(\vec{b}\).

At the top of the control window, locate the slider labeled a. This slider controls the angle \(\theta\) between \(\vec{a}\) and \(\vec{b}\).

As the angle a changes, observe the correspondence between the magnitude of their cross product \(|\vec{a} \times \vec{b}|\) (green), and the area of the parallelogram with sides \(\vec{a}\) and \(\vec{b}\).

Use the magnitude of \(\vec{a} \times \vec{b}\) to estimate the angle \(\theta \in [0,\pi]\) where the area of the parallelogram is maximized.

Confirm your estimate using the geometric definition of the magnitude of the cross product to identify the exact angle \(\theta\) at which the area is maximized:

\[|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta\]

4. Challenge — Construct Example

Find two vectors \(\vec{F}\) and \(\vec{G}\) in the \(xy\)-plane so that:

\[| \vec{F} \times \vec{G} | = 4\]

EXERCISE 2 — The Direction of the Cross Product & Normal Vectors

Background & Notes

Another key geometric property of the cross product is that it can be used to construct a normal vector to a plane.

(A normal vector is a vector that is perpendicular to the plane.)

If \(\vec{a}, \vec{b} \neq \vec{0}\) are vectors in 3-space that are not parallel, the direction of \(\vec{a} \times \vec{b}\) is normal (perpendicular to) the plane spanned by \(\vec{a}\) and \(\vec{b}\).

In this exercise:

• \(\vec{v}\) and \(\vec{w}\) span the plane \(x + y + z = 0\).

Assumptions

Let \(\vec{v}\) and \(\vec{w}\) be the 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. Graph (example)

Return to graph 1, and use it to estimate the angle between the cross product \(\vec{v} \times \vec{w}\) (green) and:

• The vector \(\vec{v}\).
• The vector \(\vec{w}\).
• The plane spanned by \(\vec{v}\) and \(\vec{w}\).

2. Compute (example)

Edit and run this code to compute the cross product \(\vec{v} \times \vec{w}\).

Then, use the dot product to show that \(\vec{v} \times \vec{w}\) is orthogonal to \(\vec{v}\).

Repeat to show \(\vec{v} \times \vec{w}\) is orthogonal to \(\vec{w}\).

Due to a property called linearity, showing that the cross product \(\vec{v} \times \vec{w}\) is orthogonal to both \(\vec{v}\) and \(\vec{w}\) also shows that \(\vec{v} \times \vec{w}\) is normal (perpendicular) to the plane spanned by \(\vec{v}\) and \(\vec{w}\).

3. Construct Example

Find a vector normal to the plane containing the origin \(O(0,0,0)\), and the points \(P(1,2,-1)\), \(Q(3,0,5)\).

4. Challenge — Generalize

Working symbolically:

Compute the cross product (by hand) \(\vec{a} \times \vec{b}\), where:

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

Then, use the dot product to show \(\vec{a} \times \vec{b}\) is orthogonal to both \(\vec{a}\) and \(\vec{b}\).

If \(\vec{a}\) and \(\vec{b}\) are non-zero and not parallel, this shows \(\vec{a} \times \vec{b}\) is normal to the plane spanned by \(\vec{a}\) and \(\vec{b}\).

EXERCISE 3 — The Direction of the Cross Product & Orientation — the Right-Hand Rule

Background & Notes

You’ve now shown that the cross product \(\vec{a} \times \vec{b}\) is a vector normal to the plane spanned by \(\vec{a}\) and \(\vec{b}\), whose magnitude corresponds to the area of the parallelogram spanned by \(\vec{a}\) and \(\vec{b}\).

But there are two vectors that satisfy these conditions — both are normal to the plane and have the same magnitude, but they have opposite directions.

Question: Which of these two vectors is the cross product \(\vec{a} \times \vec{b}\)?

Answer: The one with a right-handed orientation relative to \(\vec{a}\) and \(\vec{b}\).

This happens when the cross product satisfies the right-hand rule.

Assumptions

Suppose \(\vec{a}\) and \(\vec{b}\) are non-zero vectors in 3-space, that are not parallel.

Instructions

1. Graph (example)

Go to graph 3.

In the graph, identify the vectors \(\vec{a}\) (red) and \(\vec{b}\) (blue) in the \(xy\)-plane.

Their cross product \(\vec{a} \times \vec{b}\) (green) is parallel to the \(z\)-axis.

Use the a-slider at the top of the control window to to change the angle \(\theta\) between \(\vec{a}\) and \(\vec{b}\).

Note: The a-slider takes on values from \(0\) to \(2\pi\). The initial angle setting is a = \(\pi/4\).

2. Observe & Describe (example)

As the angle \(\theta\) between the vectors \(\vec{a}\) and \(\vec{b}\) changes, observe the direction of the cross product \(\vec{a} \times \vec{b}\) relative to the \(xy\)-plane.

Describe the difference in the relative positions of \(\vec{v}\) and \(\vec{w}\) in the plane when:

• \(\vec{v} \times \vec{w}\) points in the direction of the positive \(z\)-axis.
• \(\vec{v} \times \vec{w}\) points in the direction of the of the negative \(z\)-axis.

3. Generalize (example)

Return to graph 2 to repeat parts 1 and 2 for vectors \(\vec{a}\) (red) and \(\vec{b}\) (blue), replacing the \(xy\)-plane with the plane spanned by \(\vec{a}\) and \(\vec{b}\).

Compare this to the situation in the \(xy\)-plane. Can you use the situation in the \(xy\)-plane to clarify the general situation in 3-space?

4. Apply (example)

Go to graph 4, and identify the vectors \(\vec{v}\) (red) and \(\vec{w}\) (blue).

Is the cross product \(\vec{v} \times \vec{w}\) the bright-green vector, or the dark-green vector?

5. Challenge — Construct

Find a pair of vectors \(\vec{F}\) and \(\vec{G}\) so that:

\[\vec{F} \times \vec{G} = -4 \, \hat{k}\]

BIG IDEAS

The Magnitude of the Cross Product & Area (exercise 1)

Suppose \(\vec{v}, \vec{w} \in \mathbb{R}^3\) are non-zero, and not parallel.

The geometric definition of the magnitude of the cross product of non-zero vectors \(\vec{v}, \vec{w} \neq \vec{0}\) is:

\[|\vec{v} \times \vec{w}| = |\vec{v}| |\vec{w}| \sin \theta\]

Where \(\theta\) is the angle between \(\vec{v}\) and \(\vec{w}\) (\(0 \leq \theta \leq \pi\)).

Since \(|\vec{v}| |\vec{w}| \sin \theta\) is the area of the parallelogram spanned by \(\vec{v}\) and \(\vec{w}\), the magnitude of the cross product gives us a way to measure area.

The Direction of the Cross Product & Normal Vectors (exercise 2)

Suppose \(\vec{v}, \vec{w} \in \mathbb{R}^3\) are non-zero, and are not parallel.

Then \(\vec{v}\) and \(\vec{w}\) span a plane (2-d) in 3-space (3-d).

The direction of the cross product \(\vec{v} \times \vec{w}\) is normal (perpendicular) to the plane spanned by \(\vec{v}\) and \(\vec{w}\).

The Direction of the Cross Product & Orientation — the Right-Hand Rule (exercise 3)

Suppose \(\vec{v}, \vec{w} \in \mathbb{R}^3\) are non-zero, and are not parallel.

Then \(\vec{v}\) and \(\vec{w}\) span a plane (2-d) in 3-space (3-d).

If you fix the magnitude of a normal vector to this plane, you will find that there is always a second normal vector that has the same magnitude, but opposite direction.

This means there are two vectors that are normal to the plane spanned by \(\vec{v} \times \vec{w}\), and have the same magnitude \(|\vec{v} \times \vec{w}|\).

The direction of the cross product \(\vec{v} \times \vec{w}\) is chosen so that the set of vectors \(\{ \vec{v}, \vec{w}, \vec{v} \times \vec{w} \}\) has a right-handed orientation.

In other words: The cross product satisfies the right-hand rule.