Vector Calculus

Transcribed Lecture Notes (First 3 Pages)

Vector

  • We can shift a vector without changing direction.
  • Triangle law: For two vectors, form a triangle by placing them head-to-tail; the third side [first tail to last head] is the sum.
  • Parallelogram law: Place vectors tail to tail; the diagonal from the common tail is the sum.
Commutative:
A⃗+B⃗=B⃗+A⃗\vec{A} + \vec{B} = \vec{B} + \vec{A}
Associative:
(A⃗+B⃗)+C⃗=A⃗+(B⃗+C⃗)(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})
Subtraction:
A⃗−B⃗≡A⃗+(−1)B⃗\vec{A} - \vec{B} \equiv \vec{A} + (-1)\vec{B}
Multiply by Scalar:
k⋅A⃗=kA⃗k \cdot \vec{A} = k\vec{A}

Static Equilibrium Principle

Under static equilibrium, the resultant of the forces acting at a concurrent point equals zero:

Triangle Law Representation
F⃗1+F⃗2=−F⃗3\vec{F}_1 + \vec{F}_2 = -\vec{F}_3
Equilibrium Condition
F⃗1+F⃗2=−F⃗3\vec{F}_1 + \vec{F}_2 = -\vec{F}_3

Vectors in 3D Cartesian Coordinates

Let's place a vector A⃗\vec{A} so that its origin tail is at the origin of the Cartesian coordinate system.

Vector in Component Form
A⃗=Axe^x+Aye^y+Aze^z\vec{A} = A_x \hat{e}_x + A_y \hat{e}_y + A_z \hat{e}_z
Magnitude of Vector
∣A⃗∣=A=(Ax2+Ay2+Az2)12|\vec{A}| = A = \left(A_x^2 + A_y^2 + A_z^2\right)^{\frac{1}{2}}

If C⃗=kA⃗+k′B⃗\vec{C} = k\vec{A} + k'\vec{B}, then the components will be:

Cx=kAx+k′BxC_x = k A_x + k' B_x
Cy=kAy+k′ByC_y = k A_y + k' B_y
Cz=kAz+k′BzC_z = k A_z + k' B_z
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