Vectors

Vector

A physical quantity that has both magnitude and direction.

Scalar

A physical quantity that has magnitude only.

Magnitude

The size or length of a vector.

Direction

The orientation of a vector in space.

Vector Quantity

A quantity completely described by both magnitude and direction.

Scalar Quantity

A quantity completely described by magnitude only.

Examples of Scalars

Mass, time, distance, speed, temperature, energy, volume, area.

Examples of Vectors

Displacement, velocity, acceleration, force, momentum, weight.

Distance

A scalar quantity representing the total path travelled.

Displacement

A vector quantity representing the shortest straight-line distance from the initial to the final position.

Speed

A scalar quantity representing the rate of change of distance.

Velocity

A vector quantity representing the rate of change of displacement.

Mass

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

Weight

A vector quantity representing the gravitational force acting on an object.

Force

A vector quantity that causes an object to accelerate.

Acceleration

A vector quantity representing the rate of change of velocity.

Momentum

A vector quantity equal to mass multiplied by velocity.

Equal Vectors

Vectors having the same magnitude and the same direction.

Parallel Vectors

Vectors pointing in the same direction.

Antiparallel Vectors

Vectors pointing in opposite directions.

Negative Vectors

Vectors having equal magnitudes but exactly opposite directions.

Null Vector

A vector having zero magnitude and an indeterminate direction.

Zero Vector

A vector with zero magnitude.

Resultant

The single vector equivalent to the combined effect of two or more vectors.

Equilibrant

The vector equal in magnitude but opposite in direction to the resultant.

Vector Notation

Vectors are represented by bold letters or letters with arrows above them.

Magnitude Notation

The magnitude of a vector is represented by the symbol without the arrow.

Tail of a Vector

The starting point of a vector.

Head of a Vector

The arrowhead indicating the direction of a vector.

Arrow Length

Represents the magnitude of a vector.

Arrowhead

Represents the direction of a vector.

Cartesian Plane

A coordinate plane formed by the x-axis and y-axis.

Positive x-axis

The horizontal axis directed toward the right.

Negative x-axis

The horizontal axis directed toward the left.

Positive y-axis

The vertical axis directed upward.

Negative y-axis

The vertical axis directed downward.

Horizontal Component

The component of a vector along the x-axis.

Vertical Component

The component of a vector along the y-axis.

Vector Component

One of the perpendicular parts of a vector.

x-component

The horizontal component of a vector.

y-component

The vertical component of a vector.

Component Form

A method of expressing a vector using its x- and y-components.

Vector Components

A vector can be resolved into mutually perpendicular components.

Resolving a Vector

The process of finding the components of a vector.

Component Formula

Ax = A cos θ

Component Formula

Ay = A sin θ

Magnitude Formula

A = √(Ax² + Ay²)

Direction Formula

θ = tan⁻¹(Ay/Ax)

Reference Angle

The acute angle measured from the positive x-axis unless otherwise specified.

Quadrant I

x positive, y positive.

Quadrant II

x negative, y positive.

Quadrant III

x negative, y negative.

Quadrant IV

x positive, y negative.

Direction Convention

Directions may be expressed as North of East, South of East, North of West, or South of West.

Bearing

A method of expressing direction using compass directions.

North of East

The angle measured from east toward north.

South of East

The angle measured from east toward south.

North of West

The angle measured from west toward north.

South of West

The angle measured from west toward south.

Magnitude of a Vector

The length of the vector determined using the Pythagorean Theorem.

Direction of a Vector

The angle a vector makes with the reference axis.

Pythagorean Theorem

A² + B² = C².

Inverse Tangent Function

Used to determine the direction of a vector from its components.

Component Sign Convention

Right and up are positive; left and down are negative.

Unit Vector

A normalized vector with a magnitude of 1 that indicates only the direction of a vector.

Unit Vector Formula

 = A/|A|

Vector in Unit Vector Form

A = |A|Â

Normalized Vector

A vector whose magnitude is equal to 1.

Magnitude of a Unit Vector

1

Unit of a Unit Vector

None (dimensionless).

Unit Vector Along x-axis

î

Unit Vector Along y-axis

ĵ

Components of î

(1, 0)

Components of ĵ

(0, 1)

Positive î

Points toward the positive x-axis.

Negative î

Points toward the negative x-axis.

Positive ĵ

Points toward the positive y-axis.

Negative ĵ

Points toward the negative y-axis.

Vector in Cartesian Form

A = Axî + Ayĵ

x-component in Unit Vector Form

Axî

y-component in Unit Vector Form

Ayĵ

Cartesian Vector

A vector expressed using unit vectors î and ĵ.

Comparing Two Vectors

The process of determining the relationship between two vectors based on magnitude and direction.

Equal Vectors

Vectors with the same magnitude and the same direction.

Parallel (Like) Vectors

Vectors pointing in the same direction.

Antiparallel (Unlike) Vectors

Vectors pointing in opposite directions.

Negative (Opposite) Vectors

Vectors having the same magnitude but exactly opposite directions.

Collinear Vectors

Vectors acting along the same straight line.

Concurrent Vectors

Vectors whose lines of action intersect at a common point.

Coplanar Vectors

Vectors lying in the same plane.

Arithmetic Properties of Vectors

Mathematical rules governing vector operations.

Commutative Property

A + B = B + A

Associative Property

(A + B) + C = A + (B + C)

Scalar Multiplication

kA

Negative of a Vector

−A = (−1)A

Zero Vector

ϕ = 0A = 0

Scalar Multiplication Property

Multiplying a vector by a scalar changes its magnitude and possibly its direction.

Positive Scalar

The vector retains its original direction.

Negative Scalar

The vector reverses direction.

Magnitude After Scalar Multiplication

|kA| = |k||A|