Forensic Electricity Concepts
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Questions and Answers

What happens to the direction of a vector when it is multiplied by a negative scalar?

  • The direction doubles.
  • The direction is reversed. (correct)
  • The direction remains unchanged.
  • The direction becomes perpendicular.
  • What is the result of the dot product of two perpendicular vectors?

  • A vector quantity of one.
  • The product of their magnitudes.
  • The sum of their magnitudes.
  • A scalar quantity of zero. (correct)
  • Which property does the dot product of vectors not satisfy?

  • Nonlinear property.
  • Distributive property.
  • Commutative property.
  • Associative property. (correct)
  • What characterizes the cross product of two vectors?

    <p>Resulting in a vector quantity.</p> Signup and view all the answers

    What is the relationship between the angle and the sine function in the cross product formula?

    <p>Sine is used when vectors are perpendicular.</p> Signup and view all the answers

    What does the angle $ heta$ represent in the triangle defined by points A, B, and D?

    <p>The angle between sides P and Q at vertex D</p> Signup and view all the answers

    According to the Polygon Law of Vector Addition, what does the closing side of the polygon represent?

    <p>The resultant vector in magnitude and direction opposite to the closing side's direction</p> Signup and view all the answers

    How is the direction of the resultant vector $ar{R}$ obtained?

    <p>By writing $tan heta = rac{DC}{AD}$</p> Signup and view all the answers

    In the formula for $ar{R}$, what do the terms $ar{A}$, $ar{B}$, $ar{C}$, and $ar{D}$ symbolize?

    <p>Individual vectors with varying magnitudes</p> Signup and view all the answers

    What mathematical relationship is represented by the equation $ar{R}^2 = ar{P}^2 + ar{Q}^2 + 2 ar{P} ar{Q} cos heta$?

    <p>The relationship between the lengths of a triangle's sides</p> Signup and view all the answers

    Study Notes

    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.

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    Description

    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.

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