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Questions and Answers
What happens to the direction of a vector when it is multiplied by a negative scalar?
What is the result of the dot product of two perpendicular vectors?
Which property does the dot product of vectors not satisfy?
What characterizes the cross product of two vectors?
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What is the relationship between the angle and the sine function in the cross product formula?
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What does the angle $ heta$ represent in the triangle defined by points A, B, and D?
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According to the Polygon Law of Vector Addition, what does the closing side of the polygon represent?
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How is the direction of the resultant vector $ar{R}$ obtained?
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In the formula for $ar{R}$, what do the terms $ar{A}$, $ar{B}$, $ar{C}$, and $ar{D}$ symbolize?
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What mathematical relationship is represented by the equation $ar{R}^2 = ar{P}^2 + ar{Q}^2 + 2 ar{P} ar{Q} cos heta$?
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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.
