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Unit 2: Vectors

Physics - Class 11

This chapter explores the fundamental laws governing vector addition—Triangle, Parallelogram, and Polygon laws—along with vector resolution, scalar (dot) and vector (cross) products. Each concept is defined, illustrated with diagrams, and reinforced with worked examples relevant to Class 11 Physics.

No MCQ questions available for this chapter.

Unit 2: Vectors

Introduction to Vectors

In physics, quantities that possess both magnitude and direction are called vectors. Examples include displacement, velocity, acceleration, force, and momentum. Unlike scalars (which have only magnitude), vectors follow specific rules for addition, subtraction, and multiplication. This chapter details the graphical laws for vector addition, the analytical method of resolving vectors into components, and the two primary vector products: the scalar (dot) product and the vector (cross) product.

1. Triangle, Parallelogram and Polygon Laws

1.1 Triangle Law of Vector Addition

The Triangle Law states that if two vectors are represented as two sides of a triangle taken in order, then their resultant vector is represented by the third side of the triangle taken in the opposite order.

Triangle Law of Vector Addition
Figure 1: Triangle Law – Vectors A and B placed head‑to‑tail; resultant R closes the triangle.

Mathematically, if \(\vec{A}\) and \(\vec{B}\) are two vectors, the resultant \(\vec{R}\) is given by:

\(\vec{R} = \vec{A} + \vec{B}\)

1.2 Parallelogram Law of Vector Addition

The Parallelogram Law states that if two vectors are represented by the adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

Parallelogram Law of Vector Addition
Figure 2: Parallelogram Law – Vectors A and B as adjacent sides; resultant R along the diagonal.

The magnitude of the resultant \(R\) is derived from the law of cosines:

\(R = \sqrt{A^2 + B^2 + 2AB\cos\theta}\)
where \(A = |\vec{A}|\), \(B = |\vec{B}|\), and \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\).

1.3 Direction of the Resultant

The direction \(\alpha\) of the resultant relative to vector \(\vec{A}\) is given by:

\(\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}\)

1.4 Polygon Law of Vector Addition

The Polygon Law extends the Parallelogram Law to more than two vectors. If a number of vectors are represented in magnitude and direction by the sides of an open polygon taken in order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

Polygon Law of Vector Addition
Figure 3: Polygon Law – Vectors A, B, C, D placed head‑to‑tail; resultant R closes the polygon.

Analytically, the resultant of \(n\) vectors is the vector sum:

\(\vec{R} = \sum_{i=1}^{n} \vec{A}_i\)

2. Resolution of Vectors

Resolution is the process of splitting a single vector into two or more component vectors, usually along mutually perpendicular axes (e.g., x‑ and y‑axes). This simplifies vector addition and subtraction.

2.1 Components Along Axes

For a vector \(\vec{A}\) making an angle \(\theta\) with the positive x‑axis:

  • Horizontal component: \(A_x = A\cos\theta\)
  • Vertical component: \(A_y = A\sin\theta\)

These components are themselves vectors:

\(\vec{A}_x = A_x\,\hat{i}\), \(\vec{A}_y = A_y\,\hat{j}\)

2.2 Unit Vectors

Unit vectors have a magnitude of exactly one and indicate direction. In a Cartesian coordinate system:

  • \(\hat{i}\) – unit vector along the +x axis
  • \(\hat{j}\) – unit vector along the +y axis
  • \(\hat{k}\) – unit vector along the +z axis

Any vector in three‑dimensional space can be expressed as:

\(\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\)

2.3 Example: Resolving a Force

A force of magnitude \(F = 50\text{ N}\) acts at an angle of \(30^\circ\) above the horizontal. Find its horizontal and vertical components.

\(F_x = F\cos30^\circ = 50 \times \frac{\sqrt{3}}{2} \approx 43.3\text{ N}\)

\(F_y = F\sin30^\circ = 50 \times \frac{1}{2} = 25\text{ N}\)

Thus, \(\vec{F} = 43.3\,\hat{i} + 25\,\hat{j}\;\text{N}\).

3. Scalar and Vector Products

Vectors can be multiplied in two distinct ways: the scalar (dot) product, which yields a scalar, and the vector (cross) product, which yields another vector.

3.1 Scalar Product (Dot Product)

The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is defined as:

\(\vec{A} \cdot \vec{B} = AB\cos\theta\)
where \(A = |\vec{A}|\), \(B = |\vec{B}|\), and \(\theta\) is the angle between them.

Properties:

  • Commutative: \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\)
  • Distributive over addition: \(\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\)
  • Scalar multiplication: \((c\vec{A}) \cdot \vec{B} = c(\vec{A} \cdot \vec{B})\)

Applications:

  • Work done by a force: \(W = \vec{F} \cdot \vec{d}\)
  • Power: \(P = \vec{F} \cdot \vec{v}\)
  • Projection of one vector onto another

Example: Work Done

A force \(\vec{F} = (10\hat{i} + 5\hat{j})\text{ N}\) moves an object through a displacement \(\vec{d} = (3\hat{i} + 4\hat{j})\text{ m}\). Compute the work done.

\(W = \vec{F} \cdot \vec{d} = (10)(3) + (5)(4) = 30 + 20 = 50\text{ J}\)

3.2 Vector Product (Cross Product)

The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is a vector \(\vec{C}\) perpendicular to both \(\vec{A}\) and \(\vec{B}\), with magnitude:

\(|\vec{A} \times \vec{B}| = AB\sin\theta\)

Its direction is given by the right‑hand rule: point the fingers of your right hand in the direction of \(\vec{A}\), curl them toward \(\vec{B}\); your thumb points in the direction of \(\vec{A} \times \vec{B}\).

Properties:

  • Anti‑commutative: \(\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})\)
  • Distributive over addition: \(\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}\)
  • Scalar multiplication: \((c\vec{A}) \times \vec{B} = c(\vec{A} \times \vec{B})\)
  • Cross product of parallel vectors is zero: \(\vec{A} \times \vec{A} = \vec{0}\)

Applications:

  • Torque: \(\vec{\tau} = \vec{r} \times \vec{F}\)
  • Angular momentum: \(\vec{L} = \vec{r} \times \vec{p}\)
  • Magnetic force on a moving charge: \(\vec{F} = q\vec{v} \times \vec{B}\)

Example: Torque Calculation

A force \(\vec{F} = (0, 0, 10)\text{ N}\) is applied at a point with position vector \(\vec{r} = (2, 3, 0)\text{ m}\) relative to the origin. Find the torque about the origin.

Using the determinant method:

\(\vec{\tau} = \vec{r} \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ 2 & 3 & 0\\ 0 & 0 & 10 \end{vmatrix} = (3\cdot10 - 0\cdot0)\hat{i} - (2\cdot10 - 0\cdot0)\hat{j} + (2\cdot0 - 3\cdot0)\hat{k} = 30\hat{i} - 20\hat{j} + 0\hat{k}\;\text{N·m}\)

Thus, \(\vec{\tau} = (30\hat{i} - 20\hat{j})\text{ N·m}\).

4. Summary of Key Formulas

Concept Formula Description
Resultant Magnitude (Parallelogram/Triangle) \(R = \sqrt{A^2 + B^2 + 2AB\cos\theta}\) Magnitude of sum of two vectors
Resultant Direction \(\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}\) Angle of resultant relative to \(\vec{A}\)
Vector Components \(A_x = A\cos\theta\), \(A_y = A\sin\theta\) Resolution into x and y axes
Vector in Unit‑Vector Form \(\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\) Expression using unit vectors
Dot Product \(\vec{A} \cdot \vec{B} = AB\cos\theta\) Scalar product
Cross Product Magnitude \(|\vec{A} \times \vec{B}| = AB\sin\theta\) Vector product magnitude
Cross Product Direction Right‑hand rule Perpendicular to both vectors

5. Concluding Remarks

Mastering vector laws and operations is essential for solving problems in mechanics, electromagnetism, and many other branches of physics. The Triangle, Parallelogram, and Polygon laws provide intuitive graphical methods for addition, while resolution into components and the use of unit vectors enable precise algebraic manipulation. The scalar and vector products extend vector algebra to compute work, torque, angular momentum, and other physical quantities. Practice with the examples provided, and attempt additional problems to solidify your understanding.

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