Scalars & Vectors (Video Lecture)

A set of vocabulary flashcards covering scalars, vectors, and vector operations from the lecture notes.

Scalar

A physical quantity that has only magnitude (no direction). Examples: mass, length, time, temperature, volume, density.

Vector

A physical quantity that has both magnitude and direction. Examples: position, displacement, velocity, acceleration, force, momentum.

Magnitude

The size or length of a vector; the scalar value representing its extent.

Direction

The orientation of a vector, indicating where it points.

Displacement

A vector quantity representing the change in position from initial to final point, including magnitude and direction.

Position

A vector indicating an object's location in space.

Velocity

A vector representing the rate of change of position with direction.

Acceleration

A vector representing the rate of change of velocity.

Force

A vector quantity that can cause acceleration of a body.

Mass

A scalar quantity representing the amount of matter in an object.

Length

A scalar quantity representing a linear extent (measured in meters).

Temperature

A scalar quantity representing thermal state.

Volume

A scalar quantity representing the amount of space occupied.

Density

A scalar quantity defined as mass per unit volume.

Representation of a Vector

A vector is shown with an arrow; tail at A, head at B; the vector is written as AB (with an arrow).

Arrow Notation

Vectors are written with an arrow above a capital letter (e.g., A with an arrow).

Unit Vector

A vector with magnitude 1 in the direction of a given vector; used to indicate direction. A = |A| û.

Cartesian Unit Vectors

i, j, k are unit vectors along the x-, y-, and z-axes respectively.

i, j, k

Unit vectors along x, y, z axes used in Cartesian coordinates.

Resolution of a Vector

Splitting a vector into two or more components whose combined effect equals the original vector.

Rectangular Components 2D

A = Ax i + Ay j; components along x and y.

Rectangular Components 3D

A = Ax i + Ay j + A_z k; components along x, y, z.

Component

A projection of a vector along a given axis, e.g., Ax, Ay, A_z.

Vector Addition

Combining two vectors to produce a resultant; represented by R = A + B.

Triangle Law

A geometric method of vector addition by placing vectors head-to-tail to form a triangle.

Parallelogram Law

A geometric method of vector addition using a parallelogram; the diagonal gives the resultant.

Polygon Law

Vector addition by arranging vectors to form a polygon; the closing side is the resultant.

Commutative Property

A + B = B + A in vector addition.

Associative Property

(A + B) + C = A + (B + C) in vector addition.

Subtraction of Vectors

A − B is defined as A + (−B).

Scalar Multiplication

Multiplying a vector by a scalar scales its magnitude by |scalar|; direction remains if scalar is positive, reverses if negative.

Dot Product (Scalar Product)

A · B = AB cos θ; a scalar representing the projection of one vector onto another.

Cross Product (Vector Product)

A × B = AB sin θ; a vector perpendicular to both with magnitude AB sin θ.

Right-Hand Rule

Rule to determine the direction of A × B; point fingers along A to B, thumb points in the cross product direction.

Magnitude from Dot Product Interpretation

A · B equals the product of the magnitude of one vector and the component of the other along that vector.

Perpendicular Dot Product

If θ = 90°, A · B = 0.

Parallel Dot Product

If θ = 0°, A · B = AB (maximum positive).

Anti-Parallel Dot Product

If θ = 180°, A · B = -AB (maximum negative).

A · A

The dot product of a vector with itself equals the square of its magnitude.

Unit Vector Dot Products

î·î = ĵ·ĵ = k·k = 1; î·ĵ = ĵ·k = k·î = 0.

Cross Product Anticommutative

A × B = −(B × A).

Cross Product Distributive

A × (B + C) = A × B + A × C.

Cross Product Parallel Case

If A and B are parallel, A × B = 0.

Cross Product Perpendicular Case

If A ⟂ B, |A × B| is maximum; |A × B| = AB.

Cross Product with Itself

A × A = 0.

Cross Product with Unit Vectors

i × j = k, j × k = i, k × i = j (and reverse orders give negatives).

Magnitude from Components (2D)

For A = Ax i + Ay j, |A| = sqrt(Ax^2 + Ay^2).

Magnitude from Components (3D)

|A| = sqrt(Ax^2 + Ay^2 + A_z^2).

Dot Product in Components

A · B = Ax Bx + Ay By + Az Bz when expressed in components.

Vector Addition by Components

R = A + B has components Rx = Ax + Bx, Ry = Ay + By, Rz = Az + B_z.

Magnitude of Resultant

Length of the resultant vector; |R| = sqrt(Rx^2 + Ry^2 + R_z^2).

Direction of Resultant

The orientation of R, given by appropriate angle formulas (e.g., tan θ = Ry/Rx in 2D).