Forensic Electricity Concepts

Forensic Electricity

• A triangle is formed with points A, B, and D, where D is the apex and AB is the base.
• The adjacent side to angle D is labeled P, the opposite side as Q, and angle A is denoted as $\theta$.
• The length of segments connecting points are specified: DC, BC, AD, and BD, with the relationship $|\bar{R}|^2 = |\bar{P}|^2 + |\bar{Q}|^2 + 2|\bar{P}||\bar{Q}|cos \theta$ governing the resultant vector.
• The direction of the resultant vector $\bar{R}$ is determined by $tan\Phi = \frac{DC}{AD}$, which can also be expressed as a function of AB and BC components.

Polygon Law of Vector Addition

• States that the resultant vector of multiple vectors can be represented as the closing side of a polygon taken in order and in the opposite direction.
• A hexagon illustrates this with six vectors, emphasizing that $\bar{R} = \bar{A} + \bar{B} + \bar{C} + \bar{D}$.

Chapter 1: Mathematical Tools

Multiplication by a Scalar

• Multiplying a vector by a positive scalar changes its magnitude only, while a negative scalar reverses its direction.
• Scalar multiplication follows the distributive property: $r(\vec{A} + \vec{B}) = r\vec{A} + r\vec{B}$.

Dot Product of Two Vectors

• Defined as $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|cos\theta$, where $\theta$ is the angle between them.
• The result is a scalar quantity, termed scalar product.
• The dot product yields the maximum value when vectors are parallel and equals zero when vectors are perpendicular.

Properties of the Dot Product

• Commutative property: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$.
• Distributive property: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$.

Cross Product of Two Vectors

• Defined as $\vec{A} \times \vec{B} = |\vec{A}||\vec{B}|sin \theta \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{A}$ and $\vec{B}$.
• Produces a vector quantity; the area of the parallelogram formed by vectors A and B represents the magnitude of the cross product.

Properties of the Cross Product

• Non-commutative: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.
• Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$.

Vector Components

• Vectors can be expressed in Cartesian coordinates as $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$.
• The magnitude of vector A is calculated as $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

Triangle Law of Vector Addition

• If two vectors are represented by two sides of a triangle, their vector sum equals the third side taken in the opposite direction.
• For two vectors $\vec{P}$ and $\vec{Q}$, the triangle law can be expressed using geometric relations and the cosine rule.

Resultant Vector Derivation

• The triangle law leads to the expression $|\vec{R}|^2 = |\vec{P}|^2 + 2|\vec{P}||\vec{Q}|cos\theta + |\vec{Q}|^2$, encapsulating the relationship between vectors P, Q, and their resultant R.

Explore the fundamental concepts of Forensic Electricity through this quiz. Analyze the relationships between angles and sides in a triangle, and understand how these geometric principles contribute to forensic investigations. Test your knowledge on the application of these concepts in real-world scenarios.