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Motion in a Plane

• This chapter explores motion in two dimensions (a plane) and three dimensions (space), expanding on the previously studied one-dimensional motion.
• A simple case of motion in a plane, deals with motion with constant acceleration, including projectile and circular motion.

Scalars and Vectors

• To describe motion in two and three dimensions, you need to understand vectors.
• Vectors are required to describe quantities with both magnitude and direction.
• Physical quantities are classified as scalars or vectors based on magnitude and direction.

Scalar Quantities

• Physical quantities with only magnitude, not direction.
• Scalars are specified by a single number and a unit.
• Examples: temperature, mass, length, time, work.
• Scalar combination follows standard algebra.

Vector Quantities

• Physical quantities possessing magnitude and direction.
• Vector addition/subtraction follows triangle/parallelogram laws.
• Vectors have magnitude (a number) and direction.
• Examples: displacement, acceleration, velocity, momentum, force.
• Vectors denoted with bold face or an arrow above (e.g., F, F).
• Line length indicates magnitude, arrowhead indicates direction.
• Magnitude of a vector is its absolute value indicated by |v| = v. That is vector v has a corresponding magnitude of v

Polar Vectors

• Polar vectors have a starting or application point.
• Examples: force, displacement.

Axial Vectors

• Axial vectors represent rotational effects along the rotation axis.
• Examples: angular velocity, angular momentum, torque.
• Direction is along the axis, determined by clockwise or counterclockwise rotation.

Important Vector Definitions

Modulus of a Vector

• Modulus is the magnitude of a vector.
• Vector A has a modulus represented by |A| or A.

Unit Vector

• A vector with a magnitude of one, indicating direction.
• Unit vector of A is written as  (A cap), expressed as  = A / |A|.
• Any vector equals its magnitude times the unit vector along its direction so A = |A|Â.
• Cartesian coordinates use i, j, and k as unit vectors along the x-, y-, and z-axes, respectively.
• Unit vectors have a magnitude of one and no units or dimensions.

Null Vector

• A vector with zero magnitude, with arbitrary direction.
• Also known as the zero vector, denoted by 0.
• Example: velocity for a stationary object and acceleration for an object moving with uniform velocity.

Equal Vectors

• Two vectors having equal magnitude and the same direction.
• Vectors are "free vectors," not changing with parallel displacement.

Negative Vector

• Two vectors with equal magnitudes but opposite directions.
• The negative vector of A is represented as –A.

Collinear Vectors

• Vectors acting along the same or parallel lines.
• Collinear vectors in the same direction are parallel.
• Zero angle separating parallel vectors
• Collinear vectors in opposite directions are anti-parallel.
• 180° is the angle separating anti-parallel vectors

Coplanar Vectors

• Vectors lying in the same plane.

Co-initial Vectors

• Vectors sharing a common initial point.

Orthogonal Unit Vectors

• Two or three perpendicular unit vectors.
• Denoted by i, j, and k along the x-, y-, and z-axes.

Localized Vectors

• Vectors with a fixed initial point.
• Example: the position vector of a particle (initial point at the origin).

Non-localized Vectors

• Vectors whose initial point is not fixed, also known as free vectors
• Example: A velocity vector for particle moving along a straight line.

Position and Displacement Vectors

Position Vector

• Vector locates an object relative to coordinate system origin and denoted by r.
• Consider an object at point A at time t. The vector OA is the position vector r of the object at point A.
• Position vectors indicate minimum distance from origin and direction.

Displacement Vector

• Indicates an object's change in position over time.

• Displacement vector is a straight line from initial to final positions, independent of path.

• Consider an object with position A at time t and B at time t'. AB is the displacement vector of the object in the time interval t to t'.

• Displacement vector equals the difference of position vectors so Δr = r₂ - r₁.

• Displacement vector in two dimension Δr = (x₂ - x₁) i + (y₂ - y₁) j

• Magnitude is the square root of sum of squares so |Δr| = √((x₂ - x₁)² + (y₂ - y₁)²)

• Displacement magnitude is either less than or equal to path length.

• Magnitude in three dimensions is similar: Δr = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

Multiplication of a Vector by a Real Number

• Multiplying a vector A by a real number λ yields a new vector with a magnitude λ times that of A. The new direction is along A.
• Multiplication by a negative real number (-λ) gives a vector with magnitude |λA| but opposite direction.
• So λ(A) = λA and -λ(A) = -λA
• Multiplying a constant velocity vector by time results in a displacement vector in the direction of the velocity vector

Resultant Vector

• A single vector producing the same effect as two or more vectors.

Vectors Acting in the Same Direction

• Resultant has direction of vectors and magnitude equals vectors sum.
• If vectors A and B are acting in the same direction the Resultant vector R = A + B

Vectors Acting in Mutually Opposite Directions

• Direction is that of vector with larger magnitude; magnitude is difference.
• If vectors A and B are acting in opposite directions then Resultant vector R = A - B If |B| > |A|, then the direction of R is along B. If |A| > |B|, then the direction of R is along A.

Conditions for Zero Resultant Vector

Triangle Law of Vector Addition

• If three vectors acting at a point are represented by a closed triangle sides, the resultant is zero.
• Object is then in equilibrium. If vectors A, B, and C act at the the same time and are represented by OP, PQ, and QO, their resultant is zero.
• If multiple vectors on an object are represented by sides of a closed polygon, the resultant vector is zero so the object is in equilibrium. Equilibrium conditions
• No linear motion.
• No rotational motion.
• Minimum potential energy for stability.

Addition of Vectors (Graphical Method)

• Vectors can be added if they have the same nature.
• A displacement vector can be added to another displacement vector only
• Graphical addition aids visualization.

Triangle Law of Vector Addition

• If two vectors are represented by two triangle sides (magnitude and direction), the resultant equals the third side in the opposite order.
• R* = A + B

Parallelogram Law of Vector Addition

• If two vectors acting on a particle are represented by two adjacent sides of a parallelogram, the resultant is the diagonal from the same point.
• R* = A + B

Polygon Law of Vector Addition

• When multiple vectors can be represented both in magnitude and direction by the sides of an open polygon taken in an order, then their resultant is represented in both magnitude and direction by the closing side of the polygon taken in opposite order.

Properties of Addition of Vectors

• Commutative so A + B = B + A
• Associative so (A+B)+C = A + (B+C)
• Distributive so λ(A+B) = λA+λB
• A + 0 = A

Properties of Subtraction of Vectors

• Subtraction of vectors doesn't follow the commutative law.
• A*-B ≠ B-A
• Subtraction of vectors doesn't associative law.
• A*-(B-C) ≠ (A - B) - C
• It follows distributive law so λ(A - B) = λA – λB

Resolution of Vectors in a Plane

• Process of splitting vectors into components in different directions that have collectively the same effect.
• Splitting a single vector results in component vectors.
• Two perpendicular components: r = xi+yj Resolution of a Space Vector (In Three Dimensions)
• In three dimensions A = Aₓi+Aᵧj+A₂k.
• The scalars x and y are components or resolved parts of 7 in the directions of the x and y axis respectively.
• l, m, n are direction cosines (cosines of the angles α, β, and γ the vector A makes with the x, y, z axes, respectively).
• l = Aₓ/A ; l² + m² + n² = 1;
• m = Aᵧ/A
• n = A₂/A

Dot Product or Scalar Product

• It is the product of the magnitudes of vectors A and B and the cosine of the angle θ between them, equals to the vector projection length so A · B = AB cos θ, equals A B cos θ or B A cos θ
• Parallel vectors have dot product A·B = AB.
• Perpendicular vectors have dot product A·B = 0.
• Anti-parallel vectors have dot product A·B = -AB therefore r=√(x2+y2).

Vector Product or Cross Product

• It is defined as the product of the magnitudes of vectors A and B and the sine of the angle θ between them.
• Given by A x B = AB sin θ n
• Where n is a unit vector in perpendicular direction
• Cross product produces a zero if two parallel or anti-parallel vectors.
• Given by formula like determinant
• i* (**a₂b₃ - a₃b₂) - j(a₁b₃ - a₃b₁) + k (a₁b₂-a₂b₁)