Question

What is the sum of cross-products?

Answer 1

The cross product of vector:

Cross product is an operation on two vectors. The resultant cross-product of two vectors is also a vector that is perpendicular to both vectors.

The formula of cross product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\overrightarrow{a}\times \overrightarrow{b}=\left|a\right|\left|b\right|\sin \left(\theta \right)n$

where $\theta$ is the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, $n$ is a unit vector perpendicular to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$ .

Sum of cross-products:

The sum of two cross products is also a vector. It is a property of cross product.

Assume that $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ and $\overrightarrow{d}$ are for vaectors.

Then $\left(\overrightarrow{a}\times \overrightarrow{b}\right)+\left(\overrightarrow{c}\times \overrightarrow{d}\right)=\left(\overrightarrow{a}-\overrightarrow{c}\right)\times \left(\overrightarrow{b}-\overrightarrow{d}\right)+\left(\overrightarrow{a}\times \overrightarrow{d}\right)+\left(\overrightarrow{c}\times \overrightarrow{b}\right)$.