Physics: Scalars and Vectors

• Physics is a natural science studying matter, its motion through space and time, and related concepts like energy and force.
• It is a fundamental scientific discipline.
• The goal of physics is to understand the behavior of the universe.

Scalars

• A scalar is a physical quantity possessing only magnitude.
• It is fully described with a numerical value and a unit .
• Scalars are added using ordinary algebra.
• Examples of scalar quantities: length, mass, time, temperature, speed, energy, and electric charge.
• Magnitude refers to the size or amount of the quantity.
• Units must be included when specifying a scalar to give the magnitude meaning.
• For example, "the mass is 10 kg" is complete, but "the mass is 10" is incomplete.

Vectors

• A vector is a physical quantity that has both magnitude and direction.
• Direction is an important attribute for vectors.
• Vectors are added using rules like the parallelogram law or triangle law.
• Examples of vector quantities: displacement, velocity, acceleration, force, weight, momentum, and electric field.
• Vectors are graphically represented as arrows; the length is proportional to the magnitude, and the arrowhead indicates direction.
• Notation for vectors includes boldface (e.g., A) or an arrow over the symbol (e.g., (\vec{A})).

Vector Components

• A vector can be resolved into components along coordinate axes.
• In a 2D Cartesian coordinate system, a vector (\vec{A}) can be expressed as (\vec{A} = A_x\hat{i} + A_y\hat{j}), where (\hat{i}) and (\hat{j}) are unit vectors along the x and y axes.
• Component magnitudes are given by (A_x = A\cos\theta) and (A_y = A\sin\theta), where (A) is the magnitude of (\vec{A}) and (\theta) is the angle between the vector and the x-axis.
• The magnitude of vector (\vec{A}) is found using the Pythagorean theorem: (A = \sqrt{A_x^2 + A_y^2}).
• The direction (\theta) of vector (\vec{A}) with respect to the x-axis is found using the inverse tangent function: (\theta = \tan^{-1}(\frac{A_y}{A_x})).

Vector Addition

• Vector addition combines two or more vectors into a vector sum.
• Vectors are added geometrically, considering both magnitude and direction.
• If (\vec{A} = A_x\hat{i} + A_y\hat{j}) and (\vec{B} = B_x\hat{i} + B_y\hat{j}), their sum (\vec{C} = \vec{A} + \vec{B}) is (\vec{C} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}).
• The magnitude of (\vec{C}) is (C = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2}), and its direction (\theta) is (\theta = \tan^{-1}(\frac{A_y + B_y}{A_x + B_x})).

Vector Subtraction

• Vector subtraction finds the difference between two vectors and is similar to vector addition.
• To subtract vector (\vec{B}) from vector (\vec{A}), add the negative of vector (\vec{B}) to vector (\vec{A}): (\vec{A} - \vec{B} = \vec{A} + (-\vec{B})).
• If (\vec{A} = A_x\hat{i} + A_y\hat{j}) and (\vec{B} = B_x\hat{i} + B_y\hat{j}), then (\vec{D} = \vec{A} - \vec{B}) is given by (\vec{D} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}).
• The magnitude of (\vec{D}) is (D = \sqrt{(A_x - B_x)^2 + (A_y - B_y)^2}), and its direction (\theta) is (\theta = \tan^{-1}(\frac{A_y - B_y}{A_x - B_x})).

Scalar (Dot) Product

• The scalar product (or dot product) of two vectors (\vec{A}) and (\vec{B}) results in a scalar quantity.
• It is defined as (\vec{A} \cdot \vec{B} = AB\cos\theta), where (A) and (B) are vector magnitudes, and (\theta) is the angle between them.
• 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_xB_x + A_yB_y + A_zB_z).
• The dot product is commutative: (\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}).
• If (\vec{A}) and (\vec{B}) are perpendicular, then (\vec{A} \cdot \vec{B} = 0).
• If (\vec{A}) and (\vec{B}) are parallel, then (\vec{A} \cdot \vec{B} = AB).

Vector (Cross) Product

• The vector product (or cross product) of two vectors (\vec{A}) and (\vec{B}) results in a vector quantity.
• The cross product is defined as (\vec{A} \times \vec{B} = AB\sin\theta \hat{n}), where (A) and (B) are vector magnitudes, (\theta) is the angle between them, and (\hat{n}) is a unit vector perpendicular to both (\vec{A}) and (\vec{B}), pointing in the direction given by the right-hand rule.
• 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_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}).
• The cross product is anti-commutative: (\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})).
• If (\vec{A}) and (\vec{B}) are parallel or anti-parallel, then (\vec{A} \times \vec{B} = 0).
• The magnitude of (\vec{A} \times \vec{B}) gives the area of the parallelogram formed by vectors (\vec{A}) and (\vec{B}).

Unit Vectors

• A unit vector has a magnitude of 1 and specifies a direction.
• Common unit vectors are (\hat{i}), (\hat{j}), and (\hat{k}), pointing along the x, y, and z axes of a Cartesian coordinate system.
• Any vector can be expressed as a sum of its components multiplied by the corresponding unit vectors, for example, (\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}).

Coordinate systems

• Vectors are often described using coordinate systems.
• The Cartesian coordinate system is very common, as well as polar or spherical coordinate systems.
• The proper selection of coordinate system can simplify a problem.