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Vector Cross Product Calculator

Enter the inputs and click on Calculate button. Empty inputs are considered 0.

\( \vec{a}\times\vec{b} = \begin{vmatrix}\vec{x} &\vec{y} & \vec{z} \\ a_{1} & a_{2} & a_{3}\\b_{1} & b_{2} & b_{3}\end{vmatrix} \)

= \( \vec{x} \) + \( \vec{y} \) + \( \vec{z} \)
= \( \vec{x} \) + \( \vec{y} \) + \( \vec{z} \)




What is Vector Cross Product

The Vector Cross Product, is a way of multiplying two vectors to get another vector. This product is orthogonal (perpendicular) to both of the original vectors and its magnitude represents the area of the parallelogram formed by the two vectors.

Vectors in three-dimensional space can be represented as \(\vec{v} = (v_1, v_2, v_3)\), where \(v_1\), \(v_2\), and \(v_3\) are the components of the vector along the x, y, and z axes, respectively.

Steps to Calculate the Vector Cross Product

Here are the steps to calculate the vector cross product of two vectors:

  1. Identify the components of the two vectors.
  2. Apply the formula for the cross product:

    3. \( \vec{a} \times \vec{b} = \begin{vmatrix}\vec{x} &\vec{y} & \vec{z} \\ a_{1} & a_{2} & a_{3}\\b_{1} & b_{2} & b_{3}\end{vmatrix} \)\( = \left( a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \right) \)

  3. The resulting vector is orthogonal to both original vectors.

Example

Let's take an example to illustrate the process. Suppose we have two vectors a and b:

\(\vec{a} = (a_1, a_2, a_3)\) = (2, 3, 4)

\(\vec{b} = (b_1, b_2, b_3)\) = (5, 6, 7)

To find the vector cross product \(\vec{a} \times \vec{b}\), we follow these steps:

  1. Identify the components of the vectors:

    2. \(a_1 = 2, a_2 = 3, a_3 = 4\)

    \(b_1 = 5, b_2 = 6, b_3 = 7\)

  2. Apply the cross product formula:

    3. \[ \vec{a} \times \vec{b} = \left( 3 \times 7 - 4 \times 6, 4 \times 5 - 2 \times 7, 2 \times 6 - 3 \times 5 \right) \]

  3. Calculate each component:

    4. \[ \vec{a} \times \vec{b} = \left( 21 - 24, 20 - 14, 12 - 15 \right) = \left( -3, 6, -3 \right) \]

The resulting vector is \(\vec{a} \times \vec{b} = (-3, 6, -3)\).

Therefore, the vector cross product of \(\vec{a} = (2, 3, 4)\) and \(\vec{b} = (5, 6, 7)\) is \(\vec{a} \times \vec{b} = (-3, 6, -3)\).