Scalars and Vectors

Scalars and Vectors

• Physical quantities like time, temperature, mass, density, and electric charge are fully described with a single number and a unit
• Quantities with associated direction cannot be described by a single number
• Velocity is a quantity with direction, representing the speed and direction of motion
• Force, a push or pull, requires both magnitude and direction for complete description

Scalar Quantities

• Scalar quantities possess only magnitude, not direction in space
• Examples of scalar quantities are temperature, mass, time, volume, and speed

Vector Quantities

• Vector quantities have both magnitude and direction
• Examples of vector quantities are displacement, velocity, and acceleration

Calculations with Scalars

• Calculations with scalars uses ordinary arithmetic operations
• For example, 5kg + 3kg = 8kg OR 4 x 2s = 8s

Calculations with Vectors

• Vector combination requires special rules

Displacement Vector

• Displacement is the simplest vector quantity, representing the change in position of a point
• A vector that represents the displacement is called a displacement vector
• A symbol (e.g. A) with an overhead arrow usually represents a displacement vector

Combining Displacements

• A particle undergoing displacement A, followed by displacement B, results in a total displacement
• The single displacement, denoted by C, is the line from the tail of A to the head of B
• Vector C is equal to A + B

Triangle Law of Vector Addition

• Triangle law of vector addition states the vector sum, or resultant, of displacements A and B is expressed as C = A + B

Parallelogram Law of Vector Addition

• Completing a parallelogram with vectors A and B, the diagonal represents the vector sum C

Multiple Vectors

• When adding more than two vectors, represent them end-to-end successively; the sum results from the closing side of the polygon
• If a number of vectors form a closed triangle or polygon, their sum is zero

Commutative Law of Vector Addition

• The sum of vectors doesn't depend on the order in which the vectors are taken, so, A + B = B + A

Associative Law of Vector Addition

• When adding more than two vectors, they can be grouped in any order, (A + B) + C = A + (B + C)

Subtraction of Vectors

• The vector -B has equally magnitude of B but is in the opposite direction, so, B + (−B) = 0
• Finding the difference vector (D = A - B) involves adding the vector -B to vector A

Vector Rules

• Addition and subtraction rules apply to all types of vectors, like velocities and accelerations
• Only vectors of the same type can be added
• Adding two displacements or two velocities makes sense, but not a displacement and a velocity

Vector Components

• Adding vectors geometrically can be challenging
• A simpler addition technique involves components

Resolving Vector components

• A vector's components can be found be finding the projection of the vector onto coordinate system axes
• Finding the components of a vector is called resolving the vector

Unit Vectors

• A unit vector has a magnitude of 1, without units
• In an x-y coordinate system, i denotes the unit vector pointing in the positive x-direction
• j denotes the unit vector pointing in the positive y-direction
• The vectors A sub x and A sub y are scalar magnitudes of the component vectors A sub x and A sub y

Vector Component Calculations

• From the geometry of a right triangle components for A sub x and A sub y are:
• A sub x = A cos theta
• A sub y = A sin theta

Magnitude and Direction

• The magnitude of vector A = square root of (A sub x squared) + (A sub y squared)
• The vector A makes with the positive direction of the x-axis: tan θ=A sub y / A sub x

Rotation of X and Y Axis

• Rotating from the +x-axis towards the +y-axis, θ is positive
• Rotating from the x-axis toward the -y-axis, θ is negative
• The +y-axis is at 90°, the -x-axis at 180°, and the – y-axis at 270° (or -90°)

Position Vectors

• In a 3-dimensional coordinate system, 'k' is used as a third unit component, also in the positive z-axis direction
• A vector D can be expressed as: D= D sub xi + D sub yj + D sub zk

Positions of Coordinates

• If the coordinates of a point are known, position(distance from origin)
• r = xi + yj + zk

3d Vector Direction Cosines

• cos α = x / |r|
• cos β = y / |r|
• cos γ = z / |r|
• |r| = √x2 + y2 + z²*

Dot Product

• The scalar or dot product of two vectors A and B is calculated as A. B = ABcosθ
• A and B are magnitudes and their directions are at angle θ

Dot Product Characteristics

• It is commutative, A . B = ABcosθ = B .A
• A .B = 0, then either of the two vectors is a null vector or mutually perpendicular(A = 0 OR B = 0, θ=90°)
• Vectors A and B are parallel if θ = 0 or π (pi), A . B = AB
• Scalar product associative, If A and B two vectors & m, n scalars, (mA) . (nB) = mn(A. B) = mnA. B = A. mnB = nA. mB
• Distributive law of multiplication holds: A . (B + C) = A . B + A . C

Dot Product Calculations

• If theta (θ) is the angle between A & B defined as (Axi + Ayj + Azk) and (Bxi + Byj + Bz) then

cosθ = (A. B) / (|A||B|) = ( (A sub x)(B sub x) + (A sub y)(B sub y) + (A sub z)(B sub z) / √(A sub x squared) + ( A sub y squared) + (A sub z squared) + √(B sub x squared) + ( B sub y squared) + (B sub z squared))

Cross Product

• Vector or Cross product is the vector A × B = C = |A||B|siηθ ĉ
• With two vectors A and B directions are inclined at an angle θ
• Vector product or cross product being equal to vector C, direction is perpendicular to A and B
• c with the carrot is a unit vector perpendicular to the plane of A and B

Cross Product Characteristics

• It is not commutative or: A × B = -BX A
• Vector product is associative: (mA) × B = A × (mB) = m(A × B)
• Distributive law holds: A x (B + C) = A × B + A × C Vector A and Vector B are // (parallel): A × B= 0 since sin θ= 0, then the vector product vanishes

Cross Product Calculations,

• Vector product is in terms of components of i, j, and k
• Where A = A sub xi + A sub yj+ A sub zk
• Vector B = Vector B sub xi + B sub yj+ B sub zk, then A ×B =
• i* is multiplied by ( A sub y * Vector B sub z) - ( A sub z * Vector B sub y)
• j is multiplied by ( A sub x * Vector B sub z) - (Vector A sub z * Vector B sub x)
• +k* is multiplied by ( A sub x * Vector B sub y) - (Vector A sub y * Vector B sub x)

Work Equations

• Work equations show a vector in physics is Work done by a force
• If that Force is acting on a particle = displacement direction of the force then work is determined, so:
• Work is = F sub D = FDcosθ

Rate of change of Work

• df/ dt = F sub vector D
• Vector moment or torque of a force is torque= r x force

Force in a magnetic field Vector Application

• F force acting on a point charge in a magnetic field B is equal to qv × B
• where ừ = vector represents a the velocity of the charge This is a vector is normal to the plane

Force on a particle

• Lorentz force is f = the force electrical + force magnetism =qE + q(v × B) The total force is equal q = + 𝒱 xB )

Vector Differentiation

• Vectors and derivatives: describing displacement, velocity, and acceleration: A vectors derivative found when differentiating each component w/respect to the interest variable, given Vector A = AXI + AZK+ AZK derivative of the vector equal A over DT = DAXI over DT= DAZJ+DAZK

Application of Vectors

• Application of vector differentiation = describing motion where you 1st need to describe that particles position, position vector r for a certain P = vector coordinate origin to a point. position vector wrt origin = equal xi + yj+ zk