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Wize University Linear Algebra Textbook > Products of Vectors

Cross Product Properties

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Cross Product Properties

Properties

Let u⃗, v⃗, w⃗∈R3\vec{u},\ \vec{v},\ \vec{w} \in \reals^3 be vectors.
Let c∈Rc \in \reals be a scalar.
  • u⃗ ×v⃗=−(v⃗×u⃗)\vec{u}\ \times\vec{v}=-\left(\vec{v}\times\vec{u}\right)
  • c(u⃗×v⃗)  =  (cu⃗)×v⃗  =  u⃗×(cv⃗)c\left(\vec{u}\times\vec{v}\right) \ \ =\ \ \left(c\vec{u}\right)\times\vec{v} \ \ =\ \ \vec{u}\times\left(c\vec{v}\right)
  • 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}
  • (u⃗+v⃗)×w⃗ = u⃗×w⃗ + v⃗×w⃗\left(\vec{u}+\vec{v}\right)\times\vec{w} \ =\ \vec{u}\times\vec{w}\ +\ \vec{v}\times\vec{w}
  • v⃗×v⃗=0⃗\vec{v}\times\vec{v}=\vec{0}
  • v⃗×0⃗=0⃗\vec{v}\times\vec{0}=\vec{0}
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Cross Product and Angles

Like the dot product, there is another formula for the cross product involving the angle between vectors.
For vectors u⃗, v⃗∈R3\vec u, \ \vec v \in \reals^3 and angle θ\theta between u⃗\vec u and v⃗\vec v:
∥u⃗×v⃗∥=∥u⃗∥∥v⃗∥sin⁡θ\boxed{\quad \lVert \vec{u} \times \vec{v} \rVert = \lVert\vec{u} \rVert \lVert \vec{v} \rVert \sin\theta \quad}

Watch Out!
Don't forget to take the norm on the LHS!
This is a scalar equation, just like the similar dot product formula: u⃗⋅v⃗=∥u⃗∥∥v⃗∥cos⁡θ\vec u \cdot \vec v = \lVert \vec u \rVert \lVert \vec v \rVert \cos \theta

Note
If u⃗×v⃗=0⃗\vec u \times \vec v = \vec 0 then u⃗\vec u and v⃗\vec v are parallel (or one of the vectors is 0⃗\vec 0)

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Example: Cross Product and Angles

Given ∥a⃗∥=4, ∥b⃗∥=3\lVert \vec a \rVert = 4, \ \lVert \vec b \rVert = 3 and ∥b⃗×a⃗∥=6\lVert \vec b \times \vec a \rVert = 6, what is the angle between a⃗\vec a and b⃗\vec b?
Recall the formula involving the cross product and angle and rearrange:
∥a⃗×b⃗∥=∥a⃗∥∥b⃗∥sin⁡θ∥−(b⃗×a⃗)∥∥a⃗∥∥b⃗∥=sin⁡θ∣−1∣ ∥b⃗×a⃗∥∥a⃗∥∥b⃗∥=sin⁡θ6(4)(3)=sin⁡θ12=sin⁡θ\begin{aligned} \lVert \vec{a} \times \vec{b} \rVert &=\lVert\vec{a} \rVert \lVert \vec{b} \rVert \sin\theta\\[0.5em] \dfrac{\lVert -(\vec{b} \times \vec{a}) \rVert}{\lVert\vec{a} \rVert \lVert \vec{b} \rVert} &= \sin\theta\\[1em] \dfrac{\lvert -1 \rvert\ \lVert \vec{b} \times \vec{a} \rVert}{\lVert\vec{a} \rVert \lVert \vec{b} \rVert} &= \sin\theta\\[1em] \dfrac{6}{(4)(3)} &= \sin\theta\\ \dfrac{1}{2} &= \sin\theta\\ \end{aligned}



Based on the special triangle: θ=π6\boxed{\theta = \dfrac{\pi}{6}}

Note: By the CAST rule, sin⁡\sin is also positive in the 2nd quadrant (top-left), so there is another possible answer found there.

Quadrant 2 contains angles between π2\frac{\pi}{2} and π\pi. Take our "basic" solution and subtract it from π\pi to get the alternative answer:

π−π6 = 5π6\pi-\frac{\pi}{6}\ =\ \boxed{\frac{5\pi}{6}}
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Example: Cross Product Properties

Given that u⃗×v⃗=[k2−40k+2]\vec u \times \vec v= \begin{bmatrix} k^2-4\\ 0\\ \sqrt{k+2}\\ \end{bmatrix} , determine the value(s) of kk such that u⃗\vec u and v⃗\vec v are collinear.
The vectors are collinear/parallel if the cross product is the zero vector:
u⃗×v⃗=0⃗[k2−40k+2]=[000]\begin{aligned} \vec u \times \vec v &= \vec 0\\[0.5em] \begin{bmatrix} k^2-4\\ 0\\ \sqrt{k+2}\\ \end{bmatrix} &= \begin{bmatrix} 0\\ 0\\ 0\\ \end{bmatrix} \end{aligned}
Equation 1:
k2−4=0k2=4k=±2\begin{aligned} k^2 - 4 &= 0\\ k^2 &= 4\\ k &= \pm 2 \end{aligned}
Equation 2:
k+2=0k+2=0k=−2\begin{aligned} \sqrt{k+2} &= 0\\ k+2 &= 0\\ k &= -2 \end{aligned} (no new information)

Equation 3:
k+2=0k+2=0k=−2\begin{aligned} \sqrt{k+2} &= 0\\ k+2 &= 0\\ k &= -2 \end{aligned}

Therefore only k=−2\boxed{k=-2} will ensure that all components are zero.
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Practice: Cross Product Properties

Use the cross product and dot product properties to prove:
∥u⃗×v⃗∥2+(u⃗⋅v⃗)2=∥u⃗∥2∥v⃗∥2{\lVert \vec u \times \vec v \rVert}^2 + (\vec u \cdot \vec v)^2 = {\lVert \vec u \rVert}^2 {\lVert \vec v \rVert}^2

Practice: Cross Product Properties

Let u⃗, v⃗, w⃗∈R3\vec u ,\ \vec{v} ,\ \vec w \in \reals^3 be vectors.
Let c, d∈Rc,\ d \in \reals be scalars.

Select all of the following that produce a vector.
Extra Practice
Practice Question: Dot and Cross Product Properties
Practice Question: Dot and Cross Product Properties
Given that u⃗=(1,−2,0,1)\vec{u}=\left(1,-2,0,1\right) and 𝑣⃗\vec𝑣 =(0,1,2,1),=(0,1,2,1), which of the following is true?
𝑢⃗,𝑣⃗,𝑤⃗𝑢⃗ ,\vec{𝑣} , \vec𝑤 are vectors in R3ℝ^3 and 𝑐, 𝑑 are scalars. Which of the following produce(s) a vector?
i) (𝑤⃗×𝑐𝑢⃗)+𝑑𝑣⃗(\vec𝑤× 𝑐𝑢⃗ ) + 𝑑\vec𝑣
ii) (w⃗×𝑐𝑢⃗)⋅𝑑𝑣(\vec{w}×𝑐𝑢⃗ ) \cdot 𝑑𝑣
Practice Question: Properties of Dot and Cross Products
Practice Question: Properties of Dot and Cross Products
𝑢⃗,𝑣⃗,𝑤⃗𝑢⃗ ,\vec{𝑣} , \vec𝑤 are vectors in R3ℝ^3 and 𝑐, 𝑑 are scalars. Which of the following produce(s) a vector?
i) (𝑤⃗×𝑐𝑢⃗)+𝑑𝑣⃗(\vec𝑤× 𝑐𝑢⃗ ) + 𝑑\vec𝑣
Practice Question: Vector Properties
Practice Question: Vector Properties
If u⃗, v⃗, w⃗  ∈ R3\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)
Practice: Dot and Cross Product Properties
Practice: Dot and Cross Product Properties
𝑢⃗,𝑣⃗,𝑤⃗𝑢⃗ , \vec𝑣 , 𝑤⃗ are vectors in R3ℝ^3 and 𝑐, 𝑑 are scalars. Which of the following is/are true?
Practice: Cross Product (1)
Do the vectors u⃗= <3, −1, 4>, v⃗= <6, 0, 1>, and w⃗= <−1, 2, 5> \vec{u}=~<3,~-1,~4>,~\vec{v}=~<6,~0,~1>,~\text{and}~\vec{w}=~<-1,~2,~5>~lie in the same plane?
Cross Product Properties
Practice: Cross Product Properties
Let u⃗, v⃗, w⃗∈R3\vec u ,\ \vec{v} ,\ \vec w \in \reals^3 be vectors.
Let c, d∈Rc,\ d \in \reals be scalars.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 51_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$
If u⃗=<1,−2,3>\bcb{\vec{u} = \left< 1, -2, 3 \right>} and v⃗=<2,−1,3>\bcb{\vec{v} = \left< 2, -1, 3 \right>} then u⃗×v⃗\bcb{\vec{u} \times \vec{v}} is parallel to:
Additional practice problems--vector products
Additional Practice Problems--Vector Products
1. If u⃗=(−2,3,−1),  v⃗=(−1,−2,3)\vec{u}=\left(-2,3,-1\right),\ \ \vec{v}=\left(-1,-2,3\right), find
a) u⃗ ∙ v⃗\vec{u}\ \bullet\ \vec{v}
Given that u→\overrightarrow{u}, v→\overrightarrow{v}and w→\overrightarrow{w}are vectors in R3\mathbb{R}^3and kkis a scalar (constant). Which of the following operations are defined (possible)?
u→⋅(v→×w→)\overrightarrow{u}\cdot(\overrightarrow{v}\times\overrightarrow{w})
u→+(v→⋅w→)\overrightarrow{u}+(\overrightarrow{v}\cdot\overrightarrow{w})