Vectors formulae

1. Magnitude of a Vector:

If a vector \( \vec{v} \) is given by components \(V_x, V_y, V_z\) in 3D space, the magnitude of the vector is given by:
\[ | \vec{v} | = \sqrt{V_x^2 + V_y^2 + V_z^2} \]

2. Unit Vector:
The unit vector of a given vector \( \vec{v} \) is obtained by dividing the vector by its magnitude:
\[ \hat{v} = \frac{\vec{v}}{| \vec{v} |} \]

3. Addition of Vectors:
If two vectors \( \vec{v} = V_x \hat{i} + V_y \hat{j} + V_z \hat{k} \) and \( \vec{u} = U_x \hat{i} + U_y \hat{j} + U_z \hat{k} \) are given, then their sum \( \vec{w} \) is given by:
\[ \vec{w} = \vec{v} + \vec{u} = (V_x + U_x) \hat{i} + (V_y + U_y) \hat{j} + (V_z + U_z) \hat{k} \]

4. Dot Product of Vectors:
The dot product of two vectors \( \vec{v} \) and \( \vec{u} \) is given by:
\[ \vec{v} \cdot \vec{u} = V_x U_x + V_y U_y + V_z U_z \]

5. Cross Product of Vectors:
The cross product of two vectors \( \vec{v} \) and \( \vec{u} \) is given by a vector that is orthogonal to both input vectors:
\[ \vec{v} \times \vec{u} = (V_y U_z - V_z U_y) \hat{i} - (V_x U_z - V_z U_x) \hat{j} + (V_x U_y - V_y U_x) \hat{k} \]

These are some of the basic formulae related to vectors in mathematics.