# Calculating the vector product

• intenzxboi
In summary, there is only one formula for finding the cross product of two vectors, which involves computing a determinant using the coordinates of the vectors. It does not matter if one vector has a negative sign, as the formula remains the same.
intenzxboi
i have a question I'm trying to find the cross product of the vector but don't know which formula to use.

the first one is
i(...) + j(...) + k (...)
and the other one is the same as this one but has a negative sign
i(...) - j(...) + k (...)

what is the difference and which formula should i use?

There is only one formula. You compute the following (formal) determinant.

$$\hat{A}\times\hat{B} = \left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3\end{array}\right|$$

where $\hat{A}=\left<a_1,a_2,a_3\right>$ and $\hat{B}=\left<b_1,b_2,b_3\right>$.

## What is the vector product?

The vector product, also known as the cross product, is a mathematical operation that takes two vectors as inputs and produces a third vector that is perpendicular to both of the input vectors.

## How do you calculate the vector product?

The vector product is calculated by taking the determinant of a 3x3 matrix composed of the unit vectors i, j, and k and the components of the two input vectors. The resulting vector is given by the coefficients of the i, j, and k unit vectors in the determinant.

## What is the geometric interpretation of the vector product?

The vector product has several geometric interpretations, including determining the direction of a perpendicular line, finding the area of a parallelogram formed by the two input vectors, and determining the direction of a torque or moment of force.

## What is the difference between the vector product and the scalar product?

The vector product and the scalar product are both mathematical operations on vectors, but they produce different types of results. The vector product produces a vector, while the scalar product produces a scalar (a single number). Additionally, the vector product is defined as the cross product, while the scalar product is defined as the dot product.

## How is the vector product used in physics and engineering?

The vector product is an important tool in physics and engineering for calculating forces, moments, and other physical quantities. It is particularly useful in calculating torque, determining the direction of magnetic fields, and analyzing the motion of objects in three dimensions.

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