Cross Products

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Objectives:

1.: Be able to compute a cross product.
2.: Understand the geometric meaning behind the cross product and the right hand rule.
3.: Be able to compute the area of a triangle or rectangle using cross products.
4.: Be able to compute the volume of a parallelepiped using cross products.

Recap Video

You can watch watch the following video which recaps the ideas of the section.

Test your understanding with the following questions.

The cross product of two three-dimensional vectors is a numbervector .

Which of the following is true about the cross product of two nonzero vectors $\textbf {v}$ and $\textbf {w}$ (select all that apply):

$\textbf {v}\times \textbf {w} = \textbf {w} \times \textbf {v}$ $\textbf {v} \cdot (\textbf {v} \times \textbf {w}) = 0$ $\|\textbf {v} \times \textbf {w}\| = \|\textbf {w} \times \textbf {v}\|$ $\textbf {v} \times \textbf {w} = \| \textbf {v} \| \| \textbf {w} \| \sin (\theta )$ , where $\theta$ is the angle between $\textbf {v}$ and $\textbf {w}$ .

(About part (d)) The left side is a vector, the right side is a number.

Cross Product Example

Below is a video showing an example of taking cross products.

If $\textbf {v} = \left \langle 1,2,1 \right \rangle$ and $\textbf {v} = \left \langle 3,-1,1 \right \rangle$ , compute $\textbf {v} \times \textbf {w}$ .

Problems

Consider the vectors $\textbf {v} = \left \langle 1,2,0 \right \rangle$ and $\textbf {w} = \left \langle 2,3,-1 \right \rangle$ .

The right hand rule tells us that the $y$ -component of $\textbf {w} \times \textbf {v}$ will be:
positive negative zero
The vector $\textbf {w} \times \textbf {v} =\left \langle \answer {2},\answer {-1},\answer {1} \right \rangle$ .
The vector $\textbf {v} \times \textbf {v} = \left \langle \answer {0},\answer {0},\answer {0} \right \rangle$ .
The dot product $\textbf {v} \cdot (\textbf {w} \times \textbf {v}) = \answer {0}$

If $\textbf {v} = \left \langle -1,2,1 \right \rangle$ and $\textbf {w} = \left \langle 2,0,1\right \rangle$ , then what is the area of the triangle spanned by $\textbf {v}$ and $\textbf {w}$ ?

The area of the triangle is $A=\frac {1}{2}\|\textbf {v} \times \textbf {w}\|$ . The steps:

The cross product is $\textbf {v} \times \textbf {w} = \left \langle \answer {2},\answer {3},\answer {-4} \right \rangle .$
The magnitude of the cross product is $\|\textbf {v} \times \textbf {w}\| = \answer {\sqrt {29}}$
The area of the triangle is: $A=\answer {\sqrt {29}/2}$

Find the area of the parallelogram with vertices $A(1,1)$ , $B(2,3)$ , $C(3,7)$ , $D(2,5)$ . (It may be useful to note that to take the cross product of two 2D vectors, make them into 3D vectors by adding a $0$ in the $z$ -component.) $\text {The area is: } \answer {2}.$

The area is, for example $\| \overrightarrow {AB} \times \overrightarrow {AC} \|$ .

Note $\overrightarrow {AB} = \left \langle 1,2 \right \rangle$ , and $\overrightarrow {AC} = \left \langle 2, 6 \right \rangle$ . We can add a $0$ to the $z$ -component and take the cross product of the two vectors, which results in $\left \langle 0,0,\answer {2} \right \rangle$ . The magnitude is $\answer {2}$ , which is the area of the parallelogram.

If $\textbf {v} = \left \langle -1,2,1 \right \rangle$ , $\textbf {w} = \left \langle 2,0,1 \right \rangle$ , and $\textbf {u} = \left \langle 1,1,1 \right \rangle$ , then the volume of the parallelepiped spanned by the three vectors is $V = \answer {1}$ .

It is $V = |\textbf {u} \cdot (\textbf {v} \times \textbf {w})|$ .

Are the three vectors coplanar (i.e. are they all in the same plane)?

Yes No

True/False

Throughout, $\textbf {0}$ will denote the zero vector.

If $\textbf {v}$ and $\textbf {w}$ are parallel nonzero vectors, then it must be true that $\textbf {v} \times \textbf {w} = \textbf {0}$ . TrueFalse

If $\textbf {v}$ and $\textbf {w}$ are perpendicular unit vectors, then it must be true that $\textbf {v} \times \textbf {w}$ is also a unit vector. TrueFalse

If the volume of the parallelepiped spanned by three nonzero vectors $\textbf {u}$ , $\textbf {v}$ , and $\textbf {w}$ is equal to $0$ , then it must be true that the three vectors all lie in the same plane. TrueFalse

If $\textbf {v}$ and $\textbf {w}$ are nonzero vectors with $\textbf {v} \times \textbf {w} = \textbf {w} \times \textbf {v}$ , then it must be true that $\textbf {v} \cdot \textbf {w} = 0$ . TrueFalse

Optional: Volume of Parallelepiped Formula

For those who are interested, below is a video explaining how to derive the formula for the volume of a parallelepiped.