Analytical Definition of Collinear Vectors

Two vectors $\vec{a}$ and $\vec{b}$ are defined as collinear (or parallel) if there exists a non-zero scalar $\lambda$ such that:

$$\vec{b} = \lambda \vec{a}$$

[Per the definition of collinearity in Vector Algebra, vectors are collinear if they are parallel to the same line, irrespective of their magnitudes or directions.]

Step 1: Evaluating the Relationship Between Collinearity and Magnitude

Consider the magnitudes of the two vectors. If $\vec{b} = \lambda \vec{a}$, then by the property of vector magnitude:

$$|\vec{b}| = |\lambda \vec{a}| = |\lambda| |\vec{a}|$$

The condition for the two vectors to be equal in magnitude is $|\vec{b}| = |\vec{a}|$, which implies $|\lambda| = 1$. This only occurs if $\lambda = 1$ or $\lambda = -1$.

Step 2: Counter-Example Construction

To determine if the statement "Two collinear vectors are always equal in magnitude" is true, we test for a counter-example where $\lambda \neq \pm 1$.

• Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$.
• Let $\lambda = 2$.
• Define $\vec{b} = \lambda \vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$.

Calculations:

Since $\sqrt{3} \neq 2\sqrt{3}$, the vectors are collinear but their magnitudes are not equal. [By the property of scalar multiplication of vectors].

Visual Representation of Collinear Vectors with Different Magnitudes

Conclusion

The statement is False because collinear vectors are defined by parallelism (the existence of a scalar multiple), not by identity of magnitude. Only if the scalar multiple $\lambda$ is restricted to $\pm 1$ would the magnitudes be equal.

Final Solution: False

1. Definitions & Conceptual Framework

To classify a physical quantity, we must distinguish between scalars and vectors based on their mathematical properties:

• Scalar: A physical quantity characterized solely by its magnitude. It requires a numerical value and a unit of measurement. It is independent of direction [Per Scalar Field Theory].
• Vector: A physical quantity characterized by both magnitude and a specific direction in space. It must satisfy the laws of vector addition [Per Euclidean Vector Space definitions].

2. Analysis of the Physical Quantity

The quantity provided is $40\text{ watt}$.

• Magnitude: $40$
• Unit: $\text{watt (W)}$, which is the SI unit of power.
• Dimensional Analysis: Power $P$ is defined as the rate of energy transfer or work done per unit time: $P = \frac{dW}{dt}$ [Per the Definition of Power in Classical Mechanics].

3. Logical Deduction

Power is calculated as the scalar product of force $\vec{F}$ and velocity $\vec{v}$ ($P = \vec{F} \cdot \vec{v}$). Although force and velocity are vectors, the result of a dot product is always a scalar quantity [By the Properties of the Scalar Product]. Since $40\text{ watt}$ describes the rate of energy conversion without reference to a directional component, it satisfies the criteria for a scalar quantity.

4. Visual Representation of Scalar Power

Final Solution: The measure 40 watt is a scalar quantity.

1. Conceptual Definition of Velocity

In classical mechanics, velocity is defined as the rate of change of displacement with respect to time. Mathematically, it is expressed as the derivative of the position vector $\vec{r}$ with respect to time $t$:

$$\vec{v} = \frac{d\vec{r}}{dt}$$

[Per the definitions of Kinematics in Vector Algebra]

2. Analysis of Dimensional Requirements

To classify a physical quantity as either scalar or vector, we must evaluate its adherence to the following criteria:

• Scalar: A quantity defined solely by its magnitude (a real number) and unit. It is invariant under coordinate rotation.
• Vector: A quantity defined by both magnitude and a specific direction in space, satisfying the parallelogram law of vector addition.

3. Classification Logic

Velocity possesses two fundamental attributes:

1. Magnitude: The speed of the object, which is the scalar component $|\vec{v}|$.
2. Direction: The orientation of motion, denoted by the unit vector $\hat{v}$, such that $\vec{v} = |\vec{v}|\hat{v}$.

Since velocity explicitly requires direction to be fully defined—moving at $50 \, \text{km/h}$ is physically incomplete without specifying the cardinal direction (e.g., North) or vector orientation—it behaves as a directed line segment in $\mathbb{R}^3$ space.

4. Visual Representation of Velocity Vector

5. Comparison Table

Final Solution: Velocity is a vector quantity because it possesses both magnitude (speed) and a defined direction in space.

Step 1: Defining the Measurement

The given measure is $40^\circ$. This represents a magnitude of temperature or angular measurement depending on the physical context, but inherently, it signifies a scalar value.

Step 2: Theoretical Criteria for Classification

To distinguish between scalar and vector quantities, we apply the following definitions:

• Scalar Quantity: A physical quantity that is completely described by its magnitude (a numerical value along with units) and possesses no specific direction in space [Per the definition of Scalar fields in Euclidean space].
• Vector Quantity: A physical quantity that requires both magnitude and a defined direction to be completely specified. It must also satisfy the parallelogram law of vector addition [Per the axiomatic definition of Vectors in $\mathbb{R}^n$].

Step 3: Logical Analysis of the Measure

The measurement $40^\circ$ is a pure magnitude. Whether it represents an angle between two lines or a temperature on a scale (such as Celsius), it lacks an associated directional component (a vector direction such as $\hat{i}, \hat{j}, \hat{k}$ or an orientation in 3D space) required to qualify as a vector [By the principles of Scalar Algebra].

Visual Representation of a Scalar Scalar Representation

Step 4: Classification

Final Solution: The measure $40^\circ$ is classified as a Scalar quantity.

Step 1: Theoretical Definition of Coinitial Vectors

In vector algebra, two or more vectors are classified as coinitial if they originate from the same initial point [Definition: A set of vectors $\{ \vec{a}, \vec{b}, \vec{c}, ... \}$ are coinitial if there exists a point $P$ such that $P$ is the tail of all vectors in the set].

Step 2: Analysis of the Geometric Figure

Given a square $ABCD$ with vertices mapped in a 2D Cartesian plane, we identify the vectors emanating from each vertex. Based on standard conventions for a square $ABCD$ labeled in counter-clockwise order:

Step 3: Identification of Sets of Coinitial Vectors

By inspecting the vertices where two vectors share a common starting point:

Step 4: Logical Conclusion

Since the problem asks to identify the coinitial vectors in the square, we provide the sets emanating from the vertices. These pairs satisfy the requirement of sharing a common origin [Per the properties of vectors in a geometric polygon].

Final Solution: The pairs of coinitial vectors are $(\vec{AB}, \vec{AD})$, $(\vec{BA}, \vec{BC})$, $(\vec{CB}, \vec{CD})$, and $(\vec{DA}, \vec{DC})$.

Theoretical Definition of the Physical Quantity

In classical mechanics, work done ($W$) is defined as the product of the magnitude of the force applied to an object and the magnitude of the displacement in the direction of that force. Mathematically, it is expressed as the scalar product (or dot product) of the force vector ($\vec{F}$) and the displacement vector ($\vec{d}$):

$W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos(\theta)$

[Per the definition of the Dot Product in Vector Algebra: The dot product of two vectors always yields a scalar value, as $\cos(\theta)$ is a real number ranging from $-1$ to $1$.]

Step 1: Analyzing the Nature of the Quantity

• Magnitude: Work done possesses a magnitude (e.g., $50 \text{ Joules}$), which accounts for the energy transferred.
• Direction: Work does not have a direction in space. While a force vector has a direction and a displacement vector has a direction, their product—work—represents the scalar outcome of their interaction.
• Algebraic Behavior: Work can be positive, negative, or zero based on the angle $\theta$ between the force and displacement vectors. However, being negative does not imply directionality in the vector sense; it merely indicates energy transfer direction relative to the system.

Step 2: Graphical Representation of the Dot Product

Step 3: Classification Justification

Since work done $W$ is defined by the dot product of two vectors, it lacks a direction. In vector calculus, quantities defined by the projection of one vector onto another, resulting in a magnitude only, are strictly classified as Scalars.

Final Solution: Work done is a scalar quantity.

1. Conceptual Definition of Scalar and Vector Quantities

To classify the physical quantity distance, we must adhere to the definitions established in Euclidean geometry and classical mechanics:

• Scalar Quantity: A physical quantity that possesses magnitude only and does not require a specified direction for its complete description [Scalar Field Theory].
• Vector Quantity: A physical quantity that possesses both magnitude and a defined direction in space, obeying the laws of vector addition [Parallelogram Law of Vectors].

2. Analysis of Distance

Distance is defined as the total path length covered by an object moving between two points. By definition:

• Magnitude: Distance is always a non-negative real number ($\text{distance} \geq 0$).
• Directional Independence: As an object traverses a path, the direction of motion may change continuously (e.g., in curvilinear motion). Distance tracks the cumulative length of the trajectory regardless of these changes in orientation [Kinematics fundamental principle].

3. Comparative Vector/Scalar Table

4. Visual Representation of Distance vs. Displacement

5. Final Conclusion

Since distance is a measure of total length traversed between two points without any inherent requirement for a directional vector component, it is mathematically classified as a scalar quantity.

Final Solution: Distance is a scalar quantity.

Step 1: Identifying Physical Characteristics

To classify a measure as a scalar or a vector, we analyze two fundamental properties:

• Magnitude: A numerical value representing the quantity.
• Direction: A specific orientation or sense of displacement in space.

[Definition of Scalar: A physical quantity having magnitude but no specified direction.]

[Definition of Vector: A physical quantity having both magnitude and direction.]

Step 2: Analysis of the Given Quantity ($10 \text{ kg}$)

The quantity provided is $10 \text{ kg}$, which represents mass.

Step 3: Visual Representation of Scalar Classification

Step 4: Logical Conclusion

Since the measure $10 \text{ kg}$ only provides information regarding the amount of matter (magnitude) and lacks a directional component, it satisfies the necessary and sufficient conditions for a scalar quantity.

Final Solution: The measure $10 \text{ kg}$ is classified as a scalar.

Theoretical Foundation & Definitions

To evaluate the statement "Two vectors having same magnitude are collinear," we must strictly define the geometric and algebraic properties of vectors in $\mathbb{R}^3$:

• Magnitude of a Vector: The length of a vector $\vec{a}$, denoted as $|\vec{a}|$, is a scalar value representing its displacement magnitude.
• Collinear Vectors: Two vectors $\vec{a}$ and $\vec{b}$ are collinear (or parallel) if and only if there exists a non-zero scalar $k$ such that $\vec{a} = k\vec{b}$. [Per the definition of parallel/collinear vectors in Vector Algebra].

Step 1: Constructing a Counter-Example

Consider two vectors in a 2D Cartesian plane:

Let $\vec{a} = 1\hat{i} + 0\hat{j}$

Let $\vec{b} = 0\hat{i} + 1\hat{j}$

Step 2: Analysis of Magnitudes

Calculate the magnitude of $\vec{a}$:

$|\vec{a}| = \sqrt{1^2 + 0^2} = 1$ [By the Euclidean norm formula]

Calculate the magnitude of $\vec{b}$:

$|\vec{b}| = \sqrt{0^2 + 1^2} = 1$ [By the Euclidean norm formula]

Observation: $|\vec{a}| = |\vec{b}|$. The magnitudes are identical.

Step 3: Testing for Collinearity

For $\vec{a}$ and $\vec{b}$ to be collinear, there must exist a scalar $k$ such that $\vec{a} = k\vec{b}$:

$1\hat{i} + 0\hat{j} = k(0\hat{i} + 1\hat{j})$

$1\hat{i} + 0\hat{j} = 0\hat{i} + k\hat{j}$

Equating components:

• $1 = 0$ (Contradiction)
• $0 = k$

Since the component equality fails, no such scalar $k$ exists. Therefore, $\vec{a}$ and $\vec{b}$ are not collinear. [By the Principle of Vector Equivalence].

Visual Representation

Logical Conclusion

Since we have demonstrated a case where vectors possess equal magnitude but are not parallel (they are orthogonal in the provided example), the universal claim that equality of magnitude implies collinearity is false.

Final Solution: False. Two vectors having the same magnitude are not necessarily collinear. Collinearity requires the vectors to be scalar multiples of each other, which magnitude equality does not guarantee.

Step 1: Identifying the Components of the Measure

To classify the given quantity, 2 meters north-west, we must decompose it into its constituent parts based on the fundamental definitions of scalar and vector quantities in classical physics and mathematics [Per Euclidean Geometry and Vector Analysis]:

• Magnitude: The numerical value accompanied by a unit of measurement. Here, $2$ represents the distance or displacement length, and $meters$ ($m$) represents the SI unit for length.
• Direction: The specific spatial orientation specified by the term "north-west."

Step 2: Theoretical Classification Criteria

We apply the standard academic definitions to distinguish between the two classifications:

Step 3: Logical Deduction

Since the given measure, $2$ meters north-west, possesses a definite magnitude ($2$ meters) and a clearly defined direction (north-west, which represents an angle of $135^\circ$ from the positive x-axis or standard coordinate North), it satisfies the criteria for a vector quantity [By the definition of a displacement vector].

Visual Representation of the Displacement Vector

Step 4: Formal Conclusion

Because the measure indicates a displacement rather than a mere distance, and because it specifies a vector orientation, it is categorically a vector. In the context of Vector Algebra, this is a representation of a displacement vector $\vec{d}$.

Final Solution: The measure "2 meters north-west" is a Vector.

Step 1: Defining the Properties of Vectors

To analyze the statement, we must formally define the criteria for vector equality. Two vectors $\mathbf{a}$ and $\mathbf{b}$ are considered equal if and only if:

• They have the same magnitude ($|\mathbf{a}| = |\mathbf{b}|$).
• They have the same direction.

[Per the definition of Equality of Vectors in Vector Algebra].

Step 2: Analyzing Collinearity

Two vectors $\mathbf{a}$ and $\mathbf{b}$ are collinear if they are parallel to the same line, irrespective of their magnitudes and directions. Mathematically, $\mathbf{b} = k\mathbf{a}$ for some non-zero scalar $k$. [By the definition of Collinear Vectors].

If two vectors are collinear and have the same magnitude, there are two distinct possibilities for their direction:

• Case A: They point in the same direction (i.e., $k = 1$). In this case, the vectors are indeed equal.
• Case B: They point in opposite directions (i.e., $k = -1$). In this case, the vectors are negatives of each other ($\mathbf{a} = -\mathbf{b}$), not equal.

Step 3: Logical Counter-Example

Consider two vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\mathbf{a} = \hat{i}$ and $\mathbf{b} = -\hat{i}$.

• Magnitude: $|\mathbf{a}| = \sqrt{1^2} = 1$ and $|\mathbf{b}| = \sqrt{(-1)^2} = 1$. Thus, $|\mathbf{a}| = |\mathbf{b}|$. [By Euclidean Norm definition].
• Collinearity: $\mathbf{b} = -1 \cdot \mathbf{a}$. Since they satisfy the scalar multiplication condition, they are collinear.
• Equality: Since the direction of $\mathbf{a}$ (positive x-axis) is opposite to the direction of $\mathbf{b}$ (negative x-axis), $\mathbf{a} \neq \mathbf{b}$.

Step 4: Conclusion

Because there exists a subset of collinear vectors with the same magnitude that possess opposite directions (making them unequal), the universal statement "Two collinear vectors having the same magnitude are equal" is logically invalid.

Final Solution: False

Step 1: Identifying the Physical Quantity

The given value is $10^{-19}$ Coulomb. In physics and vector algebra, the unit "Coulomb" ($C$) is the SI derived unit of electric charge [Per the International System of Units (SI)].

Step 2: Defining Scalar and Vector Quantities

• Scalar Quantities: Physical quantities that are described entirely by their magnitude (a numerical value along with a unit) and do not possess a specific direction in space [Definition based on Scalar Algebra].
• Vector Quantities: Physical quantities that require both a magnitude and a specific direction to be fully characterized, and must follow the laws of vector addition [Definition based on Euclidean Vector Theory].

Step 3: Analytical Classification

Electric charge is an intrinsic property of matter. While charges are designated as "positive" or "negative," these signs denote the type of charge (polarity) rather than a direction in three-dimensional Cartesian space [Per the Principle of Conservation of Charge and Electromagnetism].

Step 4: Logical Deduction

Since the measure $10^{-19}$ Coulomb represents a magnitude of charge without associated spatial orientation (i.e., it does not change based on the coordinate system chosen to observe it), it satisfies the mathematical definition of a scalar.

Final Solution: $10^{-19}$ coulomb is a scalar quantity.

Step 1: Analyzing the Physical Quantity

The given quantity is $20\text{ m/s}^2$. To classify this as a scalar or a vector, we must decompose the expression into its constituent components: magnitude and direction.

• Unit Analysis: The unit $\text{m/s}^2$ (meters per second squared) represents the rate of change of velocity with respect to time [Per the definition of acceleration, $\vec{a} = \frac{d\vec{v}}{dt}$].
• Dimensional Analysis: The dimensions of this quantity are $[L T^{-2}]$, which corresponds uniquely to the physical quantity acceleration.

Step 2: Defining Scalar vs. Vector Criteria

According to the principles of classical mechanics and vector algebra:

Step 3: Logical Deduction

Acceleration is defined as the time rate of change of velocity. Since velocity ($\vec{v}$) is a vector quantity—possessing both magnitude (speed) and direction—its derivative with respect to time, acceleration ($\vec{a}$), inherits the property of direction [By the properties of vector differentiation].

Even though the specific numerical value $20$ represents the magnitude of the acceleration, the presence of the unit $\text{m/s}^2$ identifies the quantity as acceleration. Acceleration is a directed quantity; therefore, it is classified as a vector.

Step 4: Conclusion

Since the physical quantity acceleration necessitates a direction to fully describe its state in space, it does not satisfy the requirement of being a scalar quantity.

Final Solution: The measure $20 \text{ m/s}^2$ represents acceleration, which is a Vector quantity.

Given Variables & Initial Setup

Consider a square $ABCD$ in a 2D plane. Based on the geometric properties of a square, the vectors representing the sides and diagonals are defined by their direction and magnitude. In a square $ABCD$, let the vertices be represented as vectors from the origin or as displacement vectors between vertices.

Theoretical Definition of Equal Vectors

Two vectors $\vec{a}$ and $\vec{b}$ are defined as equal if and only if:

• They have the same magnitude ($|\vec{a}| = |\vec{b}|$).
• They have the same direction (parallel and pointing in the same orientation).

[Per the standard definition of vector equality in Euclidean space].

Step 1: Analyzing the Vectors of the Square

In square $ABCD$, the sides can be represented as follows:

• $\vec{AB}$ and $\vec{DC}$: These vectors are parallel and equal in length. Since the square has parallel opposite sides ($AB \parallel DC$), $\vec{AB} = \vec{DC}$ because they possess identical magnitude (length of the side) and direction.
• $\vec{AD}$ and $\vec{BC}$: These vectors are parallel and equal in length. Since $AD \parallel BC$, $\vec{AD} = \vec{BC}$ because they possess identical magnitude and direction.

Step 2: Verification of Equality

Step 3: Synthesis of Result

Based on the geometric properties of the square $ABCD$ provided in the coordinate plane:

1. The vector $\vec{AB}$ is equal to the vector $\vec{DC}$ as they are directed along the same axis (parallel) and have the same magnitude $a$.

2. The vector $\vec{AD}$ is equal to the vector $\vec{BC}$ as they are directed along the same axis (parallel) and have the same magnitude $a$.

Final Solution: The equal vectors in the square $ABCD$ are $\vec{AB} = \vec{DC}$ and $\vec{AD} = \vec{BC}$.

Theoretical Foundation: Scalar vs. Vector Classification

To classify a physical quantity, we must evaluate it against two primary criteria:

• Magnitude: A numerical value representing the size or extent of the quantity.
• Direction: A specific orientation in space required to define the quantity completely.

Definitions per standard physical conventions:

• Scalar Quantity: A physical quantity that is completely described by its magnitude alone. It obeys simple algebraic laws. [Per definition of Scalar fields in Euclidean space].
• Vector Quantity: A physical quantity that possesses both magnitude and direction and adheres to the parallelogram law of vector addition. [Per definition of Vector fields in Euclidean space].

Step 1: Analysis of "Time Period"

The time period represents the duration taken for a phenomenon to complete one full cycle or the interval between two events.

When measuring time period ($T$):

1. It possesses a numerical magnitude (e.g., $5 \text{ seconds}$).
2. It lacks spatial orientation. Time is treated as a scalar parameter in classical mechanics, moving monotonically forward without an inherent direction vector in three-dimensional space.

Step 2: Graphical Representation of Scalar Nature

Step 3: Logical Conclusion

Since the time period ($T \in \mathbb{R}^+$) is characterized solely by its magnitude and is independent of coordinate geometry directions ($\hat{i}, \hat{j}, \hat{k}$), it fulfills the criteria for a scalar quantity. [Based on the properties of scalar time in Newtonian physics].

Final Solution: Time period is a Scalar quantity.

Step 1: Definition of Collinear Vectors

Two vectors $\vec{a}$ and $\vec{b}$ are defined as collinear if and only if they are parallel to the same line, irrespective of their magnitudes and directions. Mathematically, two non-zero vectors $\vec{a}$ and $\vec{b}$ are collinear if there exists a non-zero scalar $\lambda$ such that:

$\vec{b} = \lambda \vec{a}$

[Per the definition of parallel/collinear vectors in Euclidean Space]

Step 2: Analysis of the Given Vectors

We are given two vectors, $\vec{a}$ and $-\vec{a}$. We seek to determine if there exists a scalar $\lambda$ that satisfies the collinearity condition.

Let $\vec{b} = -\vec{a}$. We can rewrite this expression as:

$\vec{b} = (-1) \vec{a}$

Here, the scalar $\lambda = -1$. Since $\lambda \neq 0$, the condition for collinearity is satisfied.

Step 3: Geometric Interpretation

Geometrically, the vector $-\vec{a}$ has the same magnitude as $\vec{a}$ but acts in the exact opposite direction. Because both vectors lie along the same line of support (the line passing through the origin and the terminal point of $\vec{a}$), they are inherently collinear.

Step 4: Logical Conclusion

• Since $\vec{b} = \lambda \vec{a}$ holds for $\lambda = -1$, the vectors are parallel.
• Parallel vectors, by definition, share the same support line or are parallel to the same line.
• Therefore, $\vec{a}$ and $-\vec{a}$ are collinear.

Final Solution: True. $\vec{a}$ and $-\vec{a}$ are collinear because $-\vec{a} = (-1)\vec{a}$, satisfying the condition for collinearity.

Step 1: Geometric Definition and Coordinate Assignment

To identify vectors that are collinear but not equal in a square $ABCD$, we establish a Cartesian coordinate system. Let the vertices of the square be defined as $A(0, a)$, $B(a, a)$, $C(a, 0)$, and $D(0, 0)$, where $a > 0$ represents the side length of the square. [By definition of a square in a 2D plane].

Step 2: Defining Collinear and Equal Vectors

• Collinear Vectors: Two vectors $\vec{p}$ and $\vec{q}$ are collinear if they are parallel to the same line, irrespective of their magnitudes and directions. [Per the definition of collinearity in vector space $\mathbb{R}^n$].
• Equal Vectors: Two vectors are equal if they have the same magnitude and the same direction.

Step 3: Analytical Identification

We evaluate pairs of parallel sides of the square $ABCD$:

1. Consider vector $\vec{AB}$ and vector $\vec{DC}$. Both lie on horizontal lines ($y=a$ and $y=0$ respectively). Thus, they are collinear. However, their directions are identical, and if their lengths are equal, they are equal vectors.
2. Consider vector $\vec{AD}$ and vector $\vec{BC}$. Both lie on vertical lines ($x=0$ and $x=a$ respectively).
3. To satisfy "Collinear but not equal," we look for vectors that lie on the same or parallel lines but differ in magnitude or direction.

Step 4: Logical Deduction for the Specific Question

In the standard square configuration provided in Figure 10.6:

• Vector $\vec{a} = \vec{AB}$ and vector $\vec{c} = \vec{DC}$ are parallel. Because they have the same direction and magnitude, $\vec{AB} = \vec{DC}$. Therefore, they are not "not equal."
• If we define vectors based on the sides, $\vec{AD}$ and $\vec{BC}$ have the same magnitude. $\vec{AD}$ points from $(0,a)$ to $(0,0)$, and $\vec{BC}$ points from $(a,a)$ to $(a,0)$. Thus $\vec{AD} = \vec{BC}$.
• The solution set: Vectors $\vec{a}$ and $\vec{c}$ (where $\vec{a} = \vec{AB}$ and $\vec{c} = \vec{CD}$) are collinear because they are parallel. However, since $\vec{CD} = -\vec{AB}$, they have opposite directions. Therefore, $\vec{AB} \neq \vec{CD}$.

Final Solution: The vectors $\vec{AB}$ and $\vec{CD}$ are collinear because they are parallel to the same line, but they are not equal because they point in opposite directions ($\vec{AB} = -\vec{CD}$).

Theoretical Foundation & Definition

To classify the physical quantity "force," we must apply the fundamental definitions established in classical mechanics and vector algebra:

• Scalar Quantity: A physical quantity that possesses only magnitude and is independent of direction. [Per the axioms of Euclidean scalars].
• Vector Quantity: A physical quantity that possesses both magnitude and a defined direction in space, obeying the laws of vector addition (e.g., triangle law or parallelogram law). [Per Newton’s Second Law of Motion].

Analytical Decomposition of Force ($\vec{F}$)

Force is defined by Newton's Second Law of Motion as the product of mass ($m$) and acceleration ($\vec{a}$):

$\vec{F} = m \vec{a}$

In this expression:

• $m$ represents the mass of the body, which is a scalar quantity.
• $\vec{a}$ represents the acceleration of the body, which is a vector quantity (defined as the rate of change of velocity, $\vec{v}$, with respect to time, $t$).

Since the product of a scalar and a vector yields a vector, force must inherently possess a directional component. [By the property of vector scaling].

Visual Representation of Force Vector

Conclusion and Classification

Because force requires a specific magnitude (Newton units) and a specific orientation to define its effect on a body, it satisfies the requirements of a vector.

Final Solution: Force is a vector quantity.

Given Variables & Initial Setup

To represent the displacement vector $\vec{d}$, we define the following parameters:

• Magnitude ($|\vec{d}|$): $40\text{ km}$. We shall use a scale factor where $1\text{ cm} = 10\text{ km}$, thus the vector length in the drawing is $4\text{ cm}$.
• Direction: $30^\circ$ East of North. This implies the angle is measured from the North (positive $y$-axis) towards the East (positive $x$-axis).

Step 1: Coordinate Geometry Calculation

To ensure high precision, we convert the polar representation $(r, \theta)$ to Cartesian coordinates $(x, y)$.

• The angle relative to the positive $x$-axis (East) is calculated as: $\theta_{std} = 90^\circ - 30^\circ = 60^\circ$. [By Complementary Angle Theorem]
• $x = r \cos(60^\circ) = 4 \times 0.5 = 2.0\text{ cm}$.
• $y = r \sin(60^\circ) = 4 \times \frac{\sqrt{3}}{2} \approx 4 \times 0.866 = 3.464\text{ cm}$.

Step 2: Graphical Representation (SVG)

Step 3: Verification of Vector Properties

The vector $\vec{d}$ is represented starting from the origin $(0,0)$ in the coordinate plane:

1. The tail is fixed at the origin representing the starting point.
2. The arrowhead terminates at $(2, 3.464)$ relative to the origin.
3. The magnitude is validated via the Pythagorean theorem: $\sqrt{2^2 + 3.464^2} = \sqrt{4 + 12} = \sqrt{16} = 4\text{ cm}$. [By Pythagorean Identity]

Final Solution: The displacement vector is represented as a directed line segment of length 4 units, originating at the coordinate intersection and oriented at an angle of 30° East of North, equivalent to a displacement of 40 km.