A doubt related to Cross Product of 2 Vectors

Question

We want to find the magnitude of C vector cross D vector.

Now my doubt is that should we measure the smallest angle between the 2 vectors (∝) in clockwise direction (from D vector to C vector) or in anticlockwise direction (from C vector to D vector)?

Answer 1

$$\vec C \; \times \; \vec D=|\vec C||\vec D|\hat n\sin \theta$$

where $\theta$ is measured from $\vec C$ to $\vec D$.

However, if you are just looking at the magnitude of the cross product, then it doesn't matter, since

$$|\vec C \; \times \; \vec D|=|\vec C||\vec D||\sin \theta|=|\vec C||\vec D||-\sin \theta|=|\vec C||\vec D||\sin (-\theta)|$$

Answer 2

The angle $\varphi$ in $$\vec{C} \times \vec{D} = \left\lVert\vec{C}\right\rVert \left\lVert\vec{D}\right\rVert \hat{e}_\perp \sin\varphi$$ is always measured around the unit vector $\hat{e}_\perp$ perpendicular to both vectors: $$\hat{e}_\perp = \frac{\vec{C} \times \vec{D}}{\left\lVert \vec{C} \times \vec{D} \right\rVert}$$ The angle $\varphi$ is measured from $\vec{C}$ to $\vec{D}$, with positive values counterclockwise around $\hat{e}_\perp$.

If you have a negative $\varphi$, then $-\varphi$ indicates the angle clockwise around $\hat{e}_\perp$ from $\vec{C}$ to $\vec{D}$.

Geometrically, $\vec{C} \times \vec{D}$ yields a vector whose direction is perpendicular to both $\vec{C}$ and $\vec{D}$, and whose magnitude is the area $A$ of a parallelogram with sides $\vec{C}$ and $\vec{D}$ meeting at a vertex, $$A = \frac{\left\lVert \vec{C} \times \vec{D} \right\rVert}{\left\lVert \vec{C} \right\rVert \; \left\lVert \vec{D} \right\rVert}$$

The notation above assumes $$\vec{a} = \left [ \begin{matrix} x \\ y \\ z \end{matrix} \right ], \quad \left\lVert \vec{a} \right\rVert = \sqrt{\vec{a} \cdot \vec{a}} = \sqrt{x^2 + y^2 + z^2}$$

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There is no real cross product in two dimensions, but there is an analog, exterior product or wedge product, that yields a scalar; the $\times$ operator is often used for it, too (and even the "cross product" name, as there is no real cross product in 2D and this analog is the closest equivalent). Given $$\vec{a} = \left [ \begin{matrix} a_x \\ a_y \end{matrix} \right ], \quad \vec{b} = \left [ \begin{matrix} b_x \\ b_y \end{matrix} \right ]$$ we have $$\begin{aligned} \vec{a} \cdot \vec{b} &= \left\lVert \vec{a} \right\rVert \left\lVert \vec{b} \right\rVert \cos\varphi \\ \vec{a} \times \vec{b} &= \left\lVert \vec{a} \right\rVert \left\lVert \vec{b} \right\rVert \sin\varphi \end{aligned}$$ where $\varphi$ is measured counterclockwise from $\vec{a}$ to $\vec{b}$.

If $\varphi$ is negative, then $-\varphi$ is measured clockwise from $\vec{a}$ to $\vec{b}$.