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Exercise 13

Avg difficulty 3.0

The cross product of the vectors

x⃗=(x1x2x3)y⃗=(y1y2y3)\vec{x}=\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}\qquad\vec{y}=\begin{pmatrix}y_{1}\\ y_{2}\\ y_{3}\end{pmatrix}

is the vector computed as this determinant.

x⃗×y⃗=det⁡((e⃗1e⃗2e⃗3x1x2x3y1y2y3))\vec{x}\times\vec{y}=\det(\begin{pmatrix}\vec{e}_{1}&\vec{e}_{2}&\vec{e}_{3}\\ x_{1}&x_{2}&x_{3}\\ y_{1}&y_{2}&y_{3}\end{pmatrix})

Note that the first row’s entries are vectors, the vectors from the standard basis for R3\mathbb{R}^{3}. Show that the cross product of two vectors is perpendicular to each vector.

Hefferon, Linear Algebra, Determinants, Exploration, Exercise 13

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.