Physics Scalars and Vectors Quiz

Scalars and Vectors

• Scalars: Quantities with only magnitude (e.g., mass, time, distance, speed).
• Vectors: Quantities with both magnitude and direction (e.g., displacement, velocity, acceleration, force).

Vector Notation and Representation

• Vectors denoted by an alphabet with an arrow (→) over its head (e.g., velocity is denoted as V→).
• Represented by a directed line segment:
  • Length indicates magnitude.
  • Arrow indicates direction.

Vector Modulus

• Modulus represents only the magnitude of a vector.
• Denoted by enclosing the vector within two vertical parallel lines (e.g., |a|).
• The modulus of a vector 'a' is: |a| = √a² = a.

Vector Classification

• Unit Vector: Modulus equals 1 (denoted by a small letter with a caret or hat symbol).
• Null Vector: Modulus equals zero. Example: The velocity of a body vertically projected upwards at its maximum height.
• Proper Vector: Modulus is non-zero.
• Like Vectors: Same direction but different magnitudes.
• Unlike Vectors: Opposite direction and different magnitudes.
• Equal Vectors: Same magnitude and direction.
• (-ve) Vectors: Same magnitude, opposite directions.
• Co-linear Vectors: All vectors lie on the same line.
• Co-planner Vectors: All vectors lie in the same plane.
• Co-initial Vectors: All vectors have the same origin.
• Orthogonal Vectors: Vectors perpendicular to each other.

Vector Addition

• Head-and-tail rule: Joining the head of one vector with the tail of another results in the resultant vector.
• General Addition: AB + BC = AC, PB + BR = PR. You cannot add vectors if they are not head-to-tail.
• Addition of Several Vectors: The resultant is found by connecting the tail of the first vector to the head of the last vector. The resultant of vectors in a closed figure is zero.

Vector Addition Properties

• Commutative Law: a + b = b + a.
• Associative Law: (a + b) + c = a + (b + c).

Vector Subtraction

• Subtracting vector 'b' from 'a' is the same as adding -b to a (a-b = a + (-b)).
• To find -b, reverse the direction of b while keeping the same magnitude.

Vector Multiplication

• Scalar Product (Dot Product): a.b = |a| |b| cos θ = ab cos θ.
• Vector Product (Cross Product): a x b = |a| |b| sin θ n = ab sin θ n.

Scalar Product Properties

• Maximum value when θ = 0° (vectors in the same direction).
• Value is 0 when θ = 90° (vectors perpendicular).
• Dot product of a vector with itself: a.a = |a|² = a².
• Commutative: a.b = b.a.
• Dot product of unit vectors: î.î = j.j = k.k = 1; î.j = j.k = k.i = 0.

Vector Product Properties

• |a x b| = ab sin θ.
• Maximum value when θ = 90° (vectors perpendicular).
• Value is 0 when θ = 0° (vectors in the same direction).
• Cross product of a vector with itself is 0: a x a = 0.
• Anti-commutative: (a x b) = -(b x a).

Triangle Law of Vector Addition

• If two vectors are represented by sides of a triangle, their resultant is represented by the third side, but the direction is reversed.
• Resultant magnitude: c = √(a² + b² + 2ab cos θ).
• Resultant direction: α = tan^-1 (b sin θ/a + b cos θ).

Special Triangle Law Cases

• θ = 0°: Vectors in the same direction, |c| = |a| + |b|.
• θ = 180°: Vectors in opposite directions, |c| = |a| - |b|.
• θ = 90°: Vectors perpendicular, use Pythagorean theorem to find the resultant.

Resolving a Vector into Components

• A vector can be resolved into two perpendicular components.
• For a vector r = xi + yj, where x = r cos θ and y = r sin θ, the vector can be written as: r = r cos θ i + r sin θ j.

Examples

• Same Direction: A car travelling 200m East and 300m East has a net displacement of 500m East.
• Opposite Directions: A car travelling 200m East and 300m West has a net displacement of 100m West.
• Perpendicular Vectors: An object moving 10m North and 10m East has a net displacement of 14.1m.
• Angle between Vectors: Two vectors of 10m each, with an angle of 60° between them, have a resultant magnitude of 10√3m and a direction of 30°.

Test your understanding of scalars and vectors with this quiz. Dive into key concepts such as vector notation, representation, and classification. Challenge yourself to identify different types of vectors and their properties.