Cross Product (Vector Product)

Cross Product (Vector Product)

• The cross product, also known as the vector product, results in a vector.
• Magnitude of the resultant vector equals the area of the parallelogram formed by the two original vectors.
• Direction of the resultant vector is normal (perpendicular) to the plane containing the two original vectors.
• The direction is determined by the right-hand rule.

Definition Illustration

• Vectors A and B can be projected to form a parallelogram.
• If θ is the angle between A and B, the area of the parallelogram is given by |A||B|sin(θ).
• |A| represents the magnitude (absolute value) of vector A.
• |B| represents the magnitude of vector B.
• Since the cross product is a vector, its direction must be specified.
• A unit vector (aₙ) normal to the plane containing A and B indicates the direction.
• The cross product A x B is defined as |A||B|sin(θ) aₙ, where aₙ is the unit normal vector.

Cross Product in Cartesian Coordinates

• Given vectors A = Ax aₓ + Ay aᵧ + Az a₂ and B = Bx aₓ + By aᵧ + Bz a₂, their cross product can be found using a matrix determinant.
• To find A x B, arrange the unit vectors and components in a matrix:

  | aₓ  aᵧ  a₂ |
  | Ax  Ay  Az |
  | Bx  By  Bz |
  

• Shortcut:
  • For aₓ: Cover the aₓ column, multiply diagonally down (AyBz) and subtract the product diagonally up (AzBy).
  • For aᵧ: Cover the aᵧ column, multiply diagonally up (AzBx) and subtract the product diagonally down (AxBz).
  • For a₂: Cover the a₂ column, multiply diagonally down (AxBy) and subtract the product diagonally up (AyBx).
• A x B = (AyBz - AzBy) aₓ + (AzBx - AxBz) aᵧ + (AxBy - AyBx) a₂

Properties of the Cross Product

• Not Commutative: A x B ≠ B x A
• Anti-Commutative: A x B = - (B x A)
• Not Associative: A x (B x C) ≠ (A x B) x C
• Distributive: A x (B + C) = A x B + A x C

Parallel and Anti-Parallel Vectors

• A x B = |A||B|sin(θ)
• If A and B are parallel, θ = 0, so A x B = 0.
• If A and B are anti-parallel, θ = 180°, so A x B = 0.
• The cross product of two parallel or anti-parallel vectors is always zero.
• Two vectors are parallel or anti-parallel if and only if their vector product is zero.
• The cross product of any vector with itself is zero (e.g., aₓ x aₓ = 0).

Cross Product of Unit Vectors

• In a right-handed Cartesian coordinate system, the right-hand rule applies.
• Cyclic diagram: Arrange aₓ, aᵧ, and a₂ sequentially in a clockwise manner.
• For clockwise multiplication, the result is positive; for counterclockwise, the result is negative.
  • aₓ x aᵧ = a₂
  • aᵧ x a₂ = aₓ
  • a₂ x aₓ = aᵧ
  • aₓ x a₂ = -aᵧ
• Reversing the order results in the negative of the original (e.g., aᵧ x aₓ = -a₂).

Example Problem 1

• Given: A = 3aₓ + 4aᵧ + a₂ and B = 2aᵧ - 5a₂.
• Find A x B and the angle between A and B.
• Using the determinant method:

  | aₓ  aᵧ  a₂ |
  | 3   4   1  |
  | 0   2  -5  |
  

• A x B = (-20 - 2) aₓ + (0 - (-15)) aᵧ + (6 - 0) a₂ = -22aₓ + 15aᵧ + 6a₂.
• To find the angle θ:
  • |A| = √(3² + 4² + 1²) = √26.
  • |B| = √(2² + (-5)²) = √29.
  • |A x B| = √((-22)² + 15² + 6²) = √745.
  • θ = sin⁻¹(|A x B| / (|A||B|)) = sin⁻¹(√745 / (√26√29)) ≈ 83.73°.

Using Calculator in Vector Mode (Example Problem 1)

• Set calculator to vector mode.
• Define vectors A and B.
• Calculate A x B directly.
• Use the absolute value function to find |A x B|.
• Calculate the angle θ using the formula.

Example Problem 2

• Find a unit vector orthogonal to both A = aₓ - 4aᵧ + a₂ and B = 2aₓ + 3aᵧ.
• The cross product A x B is normal to both A and B.

  | aₓ  aᵧ  a₂ |
  | 1  -4   1  |
  | 2   3   0  |
  

• A x B = (0 - 3) aₓ + (2 - 0) aᵧ + (3 - (-8)) a₂ = -3aₓ + 2aᵧ + 11a₂.
• Find the unit vector:
  • |A x B| = √((-3)² + 2² + 11²) = √134.
  • Unit vector = (-3aₓ + 2aᵧ + 11a₂) / √134
  • ≈ -0.259aₓ + 0.173aᵧ + 0.95a₂.
• Check: |-0.259|² + |0.173|² + |0.95|² ≈ 1.