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Wize High School Grade 12 Calculus Textbook > Vector Products

Cross Product

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Cross Product

Given any two vectors u⃗=[u1, u2, u3]\vec{u}=\left[u_1,\ u_2,\ u_3\right] and v⃗=[v1, v2, v3]\vec{v}=\left[v_1,\ v_2,\ v_3\right] in R3R^3, the cross product (a.k.a. vector product) is defined by
u⃗×v⃗=(u2v3−u3v2, u3v1−u1v3, u1v2−u2v1)\vec{u}\times\vec{v}=\left(u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1\right)
  • The cross product between two vectors in R3R^3is a vector in R3R^3!
  • The direction of the cross product aligns with the right-hand rule


Watch Out!
We can only calculate the cross product between two vectors that are in R3R^3!

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How do we Remember This?



Example
Compute [1, 0, −3]×[2, −2, −4]\left[1,\ 0,\ -3\right]\times\left[2,\ -2,\ -4\right]
=[(0)(−4)−(−3)(−2), (−3)(2)−(1)(−4), (1)(−2)−(0)(2)]=\left[\color{orange}{\left(0\right)\left(-4\right)-\left(-3\right)\left(-2\right)}, \color{blue}{\ \left(-3\right)\left(2\right)-\left(1\right)\left(-4\right)}, \color{green}{\ \left(1\right)\left(-2\right)-\left(0\right)\left(2\right)}\right]
=[−6, −2, −2]=\left[\color{orange}{-6},\ \color{blue}{-2},\ \color{green}{-2}\right]
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Geometric Interpretation

If u⃗\vec{u} and v⃗\vec{v} are non-zero, non-parallel vectors, then u⃗×v⃗\vec{u}\times\vec{v} is a vector that is orthogonal (a.k.a. perpendicular or normal) to u⃗\vec{u} and v⃗\vec{v}


❓ What are u⃗⋅(u⃗×v⃗)\vec{u}\cdot\left(\vec{u}\times\vec{v}\right) and v⃗⋅(u⃗×v⃗)\vec{v}\cdot\left(\vec{u}\times\vec{v}\right)?
Since the cross product produces a vector that is perpendicular to the original two vectors, we know that the dot product between u⃗×v⃗\vec{u}\times\vec{v} and u⃗\vec{u} or v⃗\vec{v} is 0.


Cross Product & Angle

∣∣u⃗×v⃗∣∣=∣∣u⃗∣∣∣∣v⃗∣∣sin⁡θ\left|\left|\vec{u}\times\vec{v}\right|\right|=\left|\left|\vec{u}\right|\right|\left|\left|\vec{v}\right|\right|\sin\theta where 0≤θ≤180°0\le\theta\le180\degree is the angle between the two vectors

Practice: Cross Product

Consider u⃗=[−2, 3, −1]\vec{u}=\left[-2,\ 3,\ -1\right] and v⃗=[1, 0, −1]\vec{v}=\left[1,\ 0,\ -1\right].
How many unit vectors are perpendicular to u⃗\vec{u} and v⃗\vec{v}?

[Hint: If w⃗\vec{w} is a vector that is perpendicular to u⃗\vec{u} and v⃗\vec{v}, then any scalar multiple of w⃗\vec{w} will be a vector perpendicular to u⃗\vec{u} and v⃗\vec{v}]
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Cross Product Properties

Suppose that u⃗\vec{u}, v⃗\vec{v}, w⃗\vec{w} are vectors in R3R^3, and aa is a scalar (number).

  • Cross Product is NOT Commutative: u⃗×v⃗=−(v⃗×u⃗)\vec{u}\times\vec{v}=-\left(\vec{v}\times\vec{u}\right)
  • Distributive Law for Cross Product: u⃗×(v⃗+w⃗)=u⃗×v⃗+u⃗×w⃗\vec{u}\times\left(\vec{v}+\vec{w}\right)=\vec{u}\times\vec{v}+\vec{u}\times\vec{w}
  • Distributive Law for Scalar: a(u⃗×v⃗)=(au⃗)×v⃗=u⃗×(av⃗)a\left(\vec{u}\times\vec{v}\right)=\left(a\vec{u}\right)\times\vec{v}=\vec{u}\times\left(a\vec{v}\right)

Practice: Cross Product

Recall that i⃗=[1, 0, 0]\vec{i}=\left[1,\ 0,\ 0\right], j⃗=[0, 1, 0]\vec{j}=\left[0,\ 1,\ 0\right], and k⃗=[0, 0, 1]\vec{k}=\left[0,\ 0,\ 1\right].
Match the following cross products with the correct result.
A.
2i⃗−k⃗2\vec{i}-\vec{k}
B.
j⃗\vec{j}
C.
2i⃗2\vec{i}
D.
k⃗\vec{k}
E.
−i⃗-\vec{i}
F.
−j⃗-\vec{j}
i⃗×k⃗\vec{i}\times\vec{k}
i⃗×j⃗\vec{i}\times\vec{j}
k⃗×j⃗\vec{k}\times\vec{j}
(−i⃗)×k⃗\left(-\vec{i}\right)\times\vec{k}
k⃗×(−2j⃗)\vec{k}\times\left(-2\vec{j}\right)

j⃗×(2k⃗+i⃗)\vec{j}\times\left(2\vec{k}+\vec{i}\right)