Vectors and Equilibrium

This document provides a concise overview of key concepts related to vectors and equilibrium, covering coordinate systems, vector operations, torque, and conditions for equilibrium.

2.1 Cartesian Coordinate System

2.1.1 Describe the Cartesian coordinate system in two and three-dimensional systems;

The Cartesian coordinate system utilizes perpendicular axes to precisely define points in space.

• Two-Dimensional System:
  • Employs the x-axis (horizontal) and y-axis (vertical).
  • The intersection of these axes is the origin $(0, 0)$.
  • A point P is uniquely located by its coordinates $(a, b)$.
  • Vector direction is typically represented by the angle $\theta$ with the positive x-axis, measured counterclockwise.
• Three-Dimensional System:
  • Introduces the z-axis, which is perpendicular to both the x and y axes.
  • A point P is located by its coordinates $(a, b, c)$.
  • Vector direction in three dimensions is defined by angles with each respective axis.

2.2 Addition of Vectors by Head-to-Tail Rule

2.2.1 Explain the sum of vectors using the head-to-tail rule;

The head-to-tail rule is a graphical method for vector addition:

• Place the tail of the second vector at the head of the first vector.
• The resultant vector extends from the tail of the first vector to the head of the last vector.
• Vector addition is commutative, meaning the order does not affect the sum (e.g., $$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$).

2.2.2 Define resultant, negative, unit, null, position, and equal vectors;

• Resultant Vector: The single vector that represents the sum of two or more vectors.
• Negative Vector: A vector with the same magnitude as the original vector but pointing in the exact opposite direction.
• Unit Vector: A vector with a magnitude of 1, used solely to indicate direction.
• Null Vector (Zero Vector): A vector with zero magnitude and an arbitrary (undefined) direction.
• Position Vector: A vector that specifies the location of a point relative to a chosen origin.
• Equal Vectors: Two vectors are considered equal if they have both the same magnitude and the same direction.

2.2.3 Analyze a vector into its rectangular components;

Any vector $\vec{A}$ can be resolved into its rectangular components along the x and y axes:

Where $A$ is the magnitude of the vector and $\theta$ is the angle it makes with the positive x-axis.

2.3 Addition of Vectors by Rectangular Component Method

2.3.1 Explain the sum of vectors using perpendicular components;

This method involves resolving vectors into their components for algebraic addition:

• Resolve all vectors into their respective x and y (and z, if 3D) components.
• Find the resultant x-component ($R_x$) by summing all individual x-components.
• Find the resultant y-component ($R_y$) by summing all individual y-components.
• The magnitude of the resultant vector $R$ is given by:
  $$R = \sqrt{R_x^2 + R_y^2}$$
• The direction $\theta$ of the resultant vector is given by:
  $$\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$$

2.4 Scalar Product of Two Vectors (Dot Product)

2.4.1 Define scalar product of two vectors;

The scalar product (or dot product) of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity defined as:

Where $A$ and $B$ are the magnitudes of vectors $\vec{A}$ and $\vec{B}$ respectively, and $\theta$ is the angle between them.

2.4.2 Exemplify the scalar product of two vectors in terms of angle between them;

The value of the scalar product depends on the angle $\theta$:

• If $\theta = 0^\circ$ (vectors are parallel and in the same direction): $\vec{A} \cdot \vec{B} = AB$ (maximum positive value).
• If $\theta = 90^\circ$ (vectors are perpendicular): $\vec{A} \cdot \vec{B} = 0$.
• If $\theta = 180^\circ$ (vectors are anti-parallel): $\vec{A} \cdot \vec{B} = -AB$ (maximum negative value).

2.4.3 Describe properties of scalar product of two vectors;

• Commutative: The order of multiplication does not matter: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$.
• Parallel Vectors: If $\vec{A}$ and $\vec{B}$ are parallel, $\vec{A} \cdot \vec{B} = AB$.
• Perpendicular Vectors: If $\vec{A}$ and $\vec{B}$ are perpendicular, $\vec{A} \cdot \vec{B} = 0$.
• Self-Dot Product: The dot product of a vector with itself gives the square of its magnitude: $\vec{A} \cdot \vec{A} = A^2$.
• Component Form: In Cartesian coordinates, if $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$, then:
  $$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$$

2.5 Vector Product of Two Vectors (Cross Product)

2.5.1 Define vector product of two vectors;

The vector product (or cross product) of two vectors $\vec{A}$ and $\vec{B}$ is a vector quantity defined as:

Where $A$ and $B$ are the magnitudes of vectors $\vec{A}$ and $\vec{B}$ respectively, $\theta$ is the angle between them, and $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$.

2.5.2 Exemplify vector product of two vectors in terms of angle between them;

The magnitude of the vector product depends on the angle $\theta$:

• If $\theta = 0^\circ$ or $\theta = 180^\circ$ (vectors are parallel or anti-parallel): $\vec{A} \times \vec{B} = \vec{0}$ (null vector).
• If $\theta = 90^\circ$ (vectors are perpendicular): $|\vec{A} \times \vec{B}| = AB$ (maximum magnitude).

2.5.3 Describe properties of vector product;

• Direction: The direction of $\vec{A} \times \vec{B}$ is determined by the right-hand rule.
• Anti-commutative: The order of multiplication matters and changes the direction: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.
• Parallel Vectors: If $\vec{A}$ and $\vec{B}$ are parallel, $\vec{A} \times \vec{B} = \vec{0}$.
• Component Form: In Cartesian coordinates, if $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$, then:
  $$\vec{A} \times \vec{B} = (A_y B_z – A_z B_y) \hat{i} + (A_z B_x – A_x B_z) \hat{j} + (A_x B_y – A_y B_x) \hat{k}$$

  This can also be calculated using a determinant.

2.6 Torque

2.6.1 Describe torque as a vector product of $\vec{r} \times \vec{F}$;

Torque ($\vec{\tau}$) is the rotational equivalent of force. It is defined as the vector product of the position vector $\vec{r}$ (from the pivot point to the point where the force is applied) and the force vector $\vec{F}$:

2.6.2 Discuss applications of torque;

• Torque is the turning effect produced by a force.
• The magnitude of torque can also be expressed as:
  $$\tau = rF \sin(\theta)$$

  Where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$.

• The moment arm (or lever arm) $l$ is the perpendicular distance from the pivot to the line of action of the force, given by $l = r \sin(\theta)$. Thus, $\tau = Fl$.
• Applications: Opening a door, using a wrench to tighten a bolt, and the rotational movements in human joints are all examples of torque in action.

2.7 Equilibrium of Forces

2.7.1 Define equilibrium and its types;

Equilibrium occurs when an object is either at rest or moving with a constant velocity. In this state, the net force and the net torque acting on the object are both zero.

• Types of Equilibrium:
  • Static Equilibrium: The object is at rest (velocity is zero).
  • Dynamic Equilibrium: The object is moving with a constant velocity (acceleration is zero).

2.7.2 Describe first and second conditions of equilibrium with the help of examples from daily life;

For an object to be in complete equilibrium, both of the following conditions must be met:

• First Condition of Equilibrium (Translational Equilibrium):
  • The vector sum of all forces acting on the object is zero.
  • Mathematically: $\sum \vec{F} = 0$, which implies $\sum F_x = 0$, $\sum F_y = 0$, and $\sum F_z = 0$.
  • Example: A book resting on a table is in static translational equilibrium because the downward force of gravity is balanced by the upward normal force from the table.
• Second Condition of Equilibrium (Rotational Equilibrium):
  • The sum of all torques acting on the object about any chosen pivot point is zero.
  • Mathematically: $\sum \vec{\tau} = 0$.
  • Example: A balanced seesaw is in rotational equilibrium. The torque caused by a person on one side is balanced by the torque caused by a person on the other side, preventing rotation.