Vector Operations and Kinematics: Formulas and Concepts

Scalar Product of Vectors

The scalar product of two vectors is the number obtained by multiplying the product of their magnitudes by the cosine of the angle between them. It is represented by a dot (·) and is calculated using the formula: a · b = |a| |b| cos(α).

Condition of Perpendicularity

Two vectors are perpendicular if their scalar product is zero: a ⊥ b ↔ a · b = 0.

Angle Between Vectors

The cosine of the angle between two vectors is given by: cos(α) = (a · b) / (|a| |b|).

Vector Product of Vectors

Magnitude

The magnitude of the vector product is calculated as: |a x b| = |a| |b| sin(α).

Direction

The direction is perpendicular to the plane formed by vectors a and b.

Sense

The sense is determined by applying the right-hand rule.

Kinematics: Position, Speed, and Acceleration

Position

The position of a body relative to a reference point is defined by the vector that joins the reference point to the body's location. Its unit is the meter (m), and it can be expressed as: r = xi + yj.

Speed

Speed is the rate of change of a body's position. Its unit is meters per second (m/s). The average speed is calculated as: v = Δr / Δt. Instantaneous speed is the limit of the average speed as the time interval approaches zero. It is obtained by differentiating the position equation with respect to time.

Acceleration

Acceleration measures the rate of change of a body's velocity. It is expressed as: a = Δv / Δt. Its unit is meters per second squared (m/s²). Instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. It is obtained by differentiating the velocity equation with respect to time.

Intrinsic Components of Acceleration

Tangential Acceleration (a_t)

Tangential acceleration produces changes in the magnitude of the velocity.

• Magnitude: It is equivalent to the rate of change of the speed.
• Direction: It is tangent to the trajectory at any point.
• Sense: It is the same as the motion if the speed increases and opposite to the motion if the speed decreases.

Normal Acceleration (a_n)

Normal acceleration occurs in curvilinear motions and produces changes in the direction of the velocity.

• Magnitude: It is determined by the square of the velocity divided by the radius of the curve.
• Direction: It is radial.
• Sense: It is always directed towards the center of curvature.

Uniform Rectilinear Motion (MRU)

The equation for position is: x = x₀ + vt, where the sign (+) indicates moving away and (-) indicates moving closer.

Graphs

• Velocity vs. Time: v = v₀
• Acceleration vs. Time: a = 0
• Position vs. Time: x = x₀ + vt

Uniformly Accelerated Rectilinear Motion (MRUA)

The equation for velocity is: v = v₀ + at. The equation for position is: x = x₀ + v₀t + 1/2at².

Free Fall

• Velocity: v = gt
• Position: y = 1/2gt²

Vertical Launch Upwards

• Velocity: v = v₀ - gt
• Position: y = y₀ + v₀t - 1/2gt²