Physics

Vectors

Fundamental concepts of scalar and vector quantities. Representation and components of vectors in 2D, basic operations (vector addition and scalar multiplication), dot product, and cross product in 3D using the right-hand rule.

Scalar quantity: is a quantity that is fully described by a numerical value with its corresponding units, for example temperature T, mass m, energy E.
Vector quantity: is a quantity that \must be described by a magnitude and a direction in space. It is represented with an arrow over the symbol, or in bold letter. Ex.: velocity v⃗=v\vec{v} = \mathbf{v} ; force F⃗=F\vec{F} = \mathbf{F}
A vector v⃗\vec{v} is a directed line segment. It is designated by a pair of coordinates (vx,vy)(v_x, v_y) that indicate the horizontal and vertical displacement respectively, with respect to the R2\mathbb{R}^2 coordinate system. Ex. v⃗=(3,2)\vec{v} = (3,2)
Graphical representation of a vector in the Cartesian plane
Vector notation:
  • In Cartesian coordinates: as a pair of values, indicating the x and y position respectively of the tip of the vector. Ex.: v⃗=(4,3)\vec{v} = (4,3)
  • In unit vector notation: as a sum of unit components. The x component is designated by i^\hat{i}, and the y component by j^\hat{j}. Ex.: v⃗=4i^+3j^\vec{v} = 4\hat{i} + 3\hat{j}
  • In polar coordinates: indicating the magnitude, or size, and the direction or angle, with the horizontal. Ex.: v⃗=553∘\vec{v} = 5_{53^\circ}
Vector notation with components and angle
Vector between two points: given two points A and B, the vector AB⃗\vec{AB} is the directed line segment that indicates the displacement from the initial point A to the final point B. It is given by:
Vector between two points A and B
AB⃗=(bx−ax,by−ay)\vec{AB} = (b_x - a_x, b_y - a_y)
Magnitude ∣v⃗∣|\vec{v}|: is the size of the vector. It is represented between absolute value bars or the symbol without the top arrow, and not in bold.
Direction θ\theta: is the angle formed with the positive X semi-axis.
∣v⃗∣=v=vx2+vy2|\vec{v}| = v = \sqrt{v_x^2 + v_y^2}
θ=arctan⁡vyvx\theta = \arctan\frac{v_y}{v_x}
Calculation of magnitude and direction of a vector
Unit vector: is a vector of magnitude 1. To calculate a unit vector of v⃗\vec{v}, it is divided by its magnitude:
u^v=(vx∣v⃗∣,vy∣v⃗∣)\hat{u}_v = \left(\frac{v_x}{|\vec{v}|}, \frac{v_y}{|\vec{v}|}\right)
Resolving a vector into its components: Given a vector in polar form v⃗=v∣θ\vec{v} = v_{|\theta} we can obtain its components by applying:
vx=v⋅cos⁡θv_x = v \cdot \cos \theta
vy=v⋅sin⁡θv_y = v \cdot \sin \theta
Addition
Given u⃗=(ux,uy)\vec{u} = (u_x, u_y) and v⃗=(vx,vy)\vec{v} = (v_x, v_y). The sum will be:
u⃗+v⃗=(ux+vx,uy+vy)\vec{u} + \vec{v} = (u_x + v_x, u_y + v_y)
Example:
u⃗=(2,3)  ;  v⃗=(2,−1)⇒u⃗+v⃗=(2+2,3+(−1))=(4,2)\vec{u} = (2,3) \;;\; \vec{v} = (2,-1) \quad \Rightarrow \quad \vec{u} + \vec{v} = (2+2, 3+(-1)) = (4,2)
Graphically:
Graphical vector addition
Scalar \multiplication
Given v⃗=(vx,vy)\vec{v} = (v_x, v_y) and a scalar kk. The product k⋅v⃗k \cdot \vec{v} is:
k⋅v⃗=(k⋅vx,k⋅vy)k \cdot \vec{v} = (k \cdot v_x, k \cdot v_y)
Example: let v⃗=(2,1)  ;  k=2\vec{v} = (2,1) \;;\; k = 2
2⋅v⃗=(2⋅2,2⋅1)=(4,2)2 \cdot \vec{v} = (2 \cdot 2, 2 \cdot 1) = (4,2)
Graphically:
Scalar \multiplication graphically
Let u⃗=(ux,uy)\vec{u} = (u_x, u_y) and v⃗=(vx,vy)\vec{v} = (v_x, v_y) be two vectors. The dot product u⃗⋅v⃗\vec{u} \cdot \vec{v} is a scalar given by:
u⃗⋅v⃗=ux⋅vx+uy⋅vy\vec{u} \cdot \vec{v} = u_x \cdot v_x + u_y \cdot v_y
Alternatively, it can be calculated using the formula:
u⃗⋅v⃗=∣u⃗∣⋅∣v⃗∣⋅cos⁡θ\vec{u} \cdot \vec{v} = |\vec{u}| \cdot |\vec{v}| \cdot \cos \theta
Where θ\theta is the angle formed by the two vectors. Example:
u⃗=(2,3)  ;  v⃗=(2,−1)⇒u⃗⋅v⃗=2⋅2+3⋅(−1)=1\vec{u} = (2,3) \;;\; \vec{v} = (2,-1) \quad \Rightarrow \quad \vec{u} \cdot \vec{v} = 2 \cdot 2 + 3 \cdot (-1) = 1
Angle between two vectors:
θ=arccos⁡u⃗⋅v⃗∣u⃗∣⋅∣v⃗∣\theta = \arccos \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \cdot |\vec{v}|}
Angle between two vectors
The cross product applies to vectors in R3\mathbb{R}^3 space.
The result of the cross product (u⃗×v⃗\vec{u} \times \vec{v}) is another vector:
u⃗×v⃗=∣i^j^k^uxuyuzvxvyvz∣\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix}
with magnitude
∣u⃗×v⃗∣=∣u⃗∣⋅∣v⃗∣⋅sin⁡θ|\vec{u} \times \vec{v}| = |\vec{u}| \cdot |\vec{v}| \cdot \sin \theta
The direction of (u⃗×v⃗\vec{u} \times \vec{v}) is perpendicular to both u⃗\vec{u} and v⃗\vec{v}, and its direction follows the right-hand rule.
Right-hand rule for the cross product