Motion in a Plane: Scalar and Vector Quantities Study Notes

Scalar Quantity: A quantity that possesses magnitude only. Examples include distance, speed, time, and mass.
Vector Quantity: A quantity that has both a magnitude and a direction. Examples include displacement, velocity, acceleration, and weight.

Graphical Representation: A vector is represented by an arrow-headed straight line.
- Magnitude: The length of the arrow represents the magnitude of the vector. The distance between the initial and terminal points of a vector is called the magnitude (or length) of the vector.
- Tail: The point $O$ is called the initial point, tail, or origin.
- Head: The point $P$ is called the terminus point or head.
Co-initial Vectors: If a number of vectors lying in a plane share the same origin, they are called co-initial vectors.
Co-terminal Vectors: If a number of vectors lying in a plane share the same terminal point, they are called co-terminal vectors.
Collinear Vectors: Those vectors which act either along the same line or along parallel lines. These are divided into two types:
- Parallel Vectors: Vectors acting along the same direction. The angle between parallel vectors is $0^\circ$ .
- Anti-parallel Vectors: Vectors acting along opposite directions. The angle between anti-parallel vectors is $180^\circ$ .

Unit Vector: A vector having unit magnitude and a specific direction. For any given vector $\vec{A}$ , its unit vector $\hat{A}$ is defined as:
- $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$
Orthogonal Triad (Unit Vectors Along Three Axes): The unit vectors along the $X$ , $Y$ , and $Z$ axes are defined as $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ respectively. They are mutually perpendicular to each other.
Equal Vectors: Two vectors are considered equal if they have the same magnitude and are acting along the same direction.
Negative Vector: A vector $\vec{B}$ is said to be the negative vector of $\vec{A}$ if it has the same magnitude as $\vec{A}$ but the opposite direction ( $\vec{B} = -\vec{A}$ ).

Definition: A vector having zero magnitude and an arbitrary direction is called a null vector ( $\vec{0}$ ).
Properties of Null Vector:
- $\vec{A} + \vec{0} = \vec{A}$
- $\vec{A} - \vec{0} = \vec{A}$
- $0 \times \vec{A} = \vec{0}$
- $n \times \vec{0} = \vec{0}$
- $\vec{A} + (-\vec{A}) = \vec{0}$
Physical Meaning and Examples of Null Vectors:
- The displacement or velocity vector of a stationary object is $\vec{0}$ .
- When an object is thrown upward, its velocity at the highest point of its trajectory is $\vec{0}$ .
- When an object is undergoing uniform motion, its acceleration is a null vector.
- When an object returns to its initial starting point, its total displacement is $\vec{0}$ .

Statement: If two vectors lying in a plane can be represented both in magnitude and direction as two sides of a triangle taken in the same order, then the closing side of the triangle taken in the reverse order will give the resultant vector in both magnitude and direction.
Illustration: Let $\vec{A}$ and $\vec{B}$ be two vectors in a plane represented as sides of the triangle $OPQ$ . The closing side of $\Delta OPQ$ provides the resultant $\vec{R} = \vec{A} + \vec{B}$ .
Analytical Method for Magnitude and Direction:
- Consider vectors $\vec{A}$ and $\vec{B}$ . Let them be sides of triangle $ONQ$ where $OQ$ is the resultant $\vec{R} = \vec{A} + \vec{B}$ .
- In $\Delta PNQ$ :
  - $\sin(\theta) = \frac{NQ}{PQ} = \frac{NQ}{B} \implies NQ = B\sin(\theta)$
  - $\cos(\theta) = \frac{PN}{PQ} = \frac{PN}{B} \implies PN = B\cos(\theta)$
- In $\Delta ONQ$ , using the Pythagorean theorem where $ON = OP + PN$ :
  - $OQ^2 = ON^2 + NQ^2$
  - $OQ^2 = (OP + PN)^2 + NQ^2$
- Substituting $OQ = R$ and $OP = A$ :
  - $R^2 = (A + B\cos(\theta))^2 + (B\sin(\theta))^2$
  - $R^2 = A^2 + B^2\cos^2(\theta) + 2AB\cos(\theta) + B^2\sin^2(\theta)$
  - $R^2 = A^2 + B^2(\cos^2(\theta) + \sin^2( \theta)) + 2AB\cos(\theta)$
  - Magnitude (Law of Cosines): $R = \sqrt{A^2 + B^2 + 2AB\cos(\theta)}$
- Direction (\beta):
  - $\tan(\beta) = \frac{NQ}{ON} = \frac{NQ}{OP + PN}$
  - $\tan(\beta) = \frac{B\sin(\theta)}{A + B\cos(\theta)}$

Statement: If two vectors lying in a plane can be represented both in magnitude and direction as adjacent sides of a parallelogram starting from a point, then the diagonal starting from the same point gives the resultant vector both in magnitude and direction.
Illustration: Let $\vec{A}$ and $\vec{B}$ be adjacent sides of a parallelogram starting from a common point. The diagonal $ON$ starting from that same point represents the resultant $\vec{R} = \vec{A} + \vec{B}$ .
Analytical Method:
- Consider vectors $\vec{A}$ and $\vec{B}$ as sides of the parallelogram $OPNS$ . The diagonal $ON$ is the resultant $\vec{R} = \vec{A} + \vec{B}$ .
- In $\Delta PNQ$ :
  - $\sin(\theta) = \frac{NQ}{PQ} = \frac{NQ}{B} \implies NQ = B\sin(\theta)$
  - $\cos(\theta) = \frac{PN}{PQ} = \frac{PN}{B} \implies PN = B\cos(\theta)$
- In $\Delta ONQ$ :
  - $OQ^2 = ON^2 + NQ^2$
  - $OQ^2 = (OP + PN)^2 + NQ^2$
- Given $OQ = R$ and $OP = A$ , substituting yields:
  - $R^2 = (A + B\cos(\theta))^2 + (B\sin(\theta))^2$
  - $R^2 = A^2 + B^2(\cos^2(\theta) + \sin^2(\theta)) + 2AB\cos(\theta)$
  - Magnitude: $R = \sqrt{A^2 + B^2 + 2AB\cos(\theta)}$
- Direction:
  - $\tan(\beta) = \frac{NQ}{ON} = \frac{NQ}{OP + PN}$
  - $\tan(\beta) = \frac{B\sin(\theta)}{A + B\cos(\theta)}$

Commutative Property: Vector addition is commutative, meaning the order of addition does not matter.
- $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
Associative Property: Vector addition is associative, meaning the grouping of vectors does not change the result.
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