Unit 1: Vector Algebra and Matrices

Vector Products

The scalar product of two vectors A and B is a scalar quantity defined as:

A · B = |A| |B| cos(θ)

where |A| and |B| are the magnitudes of the vectors and θ is the angle between them.

In component form: If A = A_xî + A_yĵ + A_zk̂ and B = B_xî + B_yĵ + B_zk̂, then:
A · B = A_xB_x + A_yB_y + A_zB_z
Properties:
- It is commutative: A · B = B · A
- It is distributive: A · (B + C) = A · B + A · C
- If A and B are perpendicular (θ = 90°), A · B = 0.
- If A and B are parallel (θ = 0°), A · B = |A||B|.
- For unit vectors: î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0
Physical Application: Work done (W) by a force (F) over a displacement (d) is W = F · d.

The vector product of two vectors A and B is a vector quantity C defined as:

A × B = |A| |B| sin(θ) n̂

where n̂ is a unit vector perpendicular to the plane containing A and B, given by the Right-Hand Rule.

In component (determinant) form:

 | î ĵ k̂ | A × B = | A_x A_y A_z | | B_x B_y B_z | = (A_yB_z - A_zB_y)î - (A_xB_z - A_zB_x)ĵ + (A_xB_y - A_yB_x)k̂