Vector Calculus

Vector

• A vector is something that has both magnitude (size) and direction.
• Example:
  • Velocity → 60 km/h north
  • Force → 10 N upwards
• We usually write vectors in bold ($\mathbf{v}$) or with an arrow on top ($\vec{v}$).

Vector (Cross) Product

The vector triple product involves three vectors where one vector is crossed with the cross product of the other two.

Definition

For vectors a, b, and c:$$\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$$Result

• The result is a vector.
• It lies in the plane formed by b and c.

Important Identity (BAC–CAB rule)

$$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) – \mathbf{c}(\mathbf{a}\cdot\mathbf{b})$$Geometric meaning

It represents a vector combination of b and c, scaled by how much a aligns with them.

Scalar

• A scalar is something that has only magnitude (size), no direction.
• Example:
  • Temperature → 25°C
  • Mass → 5 kg
  • Speed → 60 km/h (just how fast, not where)

Scalar Triple Product

The scalar triple product is the dot product of one vector with the cross product of the other two.

Definition

$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$$Result

• The result is a scalar (number).

Geometric meaning

• It gives the volume of the parallelepiped formed by a, b, and c.
• If the value is zero, the vectors are coplanar.

Determinant form

\(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}\)

Product	Expression	Result	Physical meaning
Vector triple	a × (b × c)	Vector	Directional vector in plane of \(b\) and \(c\)
Scalar triple	a · (b × c)	Scalar	Volume of \(3D\) shape