Vectors and Scalars: Fundamentals

Course Outline

• Vectors are represented geometrically in 1-3 dimensions using components and direction cosines.
• Vector operations include addition, scalar multiplication, and ascertaining linear independence.
• Scalar and vector products of two vectors are considered, along with differentiation and integration with respect to a scalar variable.
• Two-dimensional coordinate geometry encompasses straight lines, circles, parabolas, ellipses, and hyperbolas.
• Analysis of tangents and normals is included.

Chapter 1: Vectors and Scalars

• Two types of physical quantities exist: scalar and vector.
• Scalar quantities are magnitude represented by a number and a unit of measurement, expressed regardless of coordinate system, like time, length, mass, density, potential, and power.
• Vectors possess both magnitude and direction, characterized by a number, direction, and unit of measurement; force, displacement, velocity, and acceleration are examples.

Representation of Vectors

• An ordered list of numbers, or "components," expresses a vector in a coordinate system.
• A vector in 2-3 dimensions would have 2-3 components.
• Vectors use bold face letters, and a directed line segment symbolizes them, with arrows pointing in the action's direction.
• The directed line segment's length, with suitable units, shows the vector's magnitude as |AB|.
• A vector V in a two-dimensional plane has initial points at the origin and a terminal point at (V1, V2), forming an ordered pair: V=<V1, V2>.
• V = (V1, V2, V3) forms an ordered triple in three-dimensional space.
• V1i, V2j, and V3k are the components of V.

Vector operations

• √(v12 + v22 + v32) denotes the magnitude of vector V.
• Given an arbitrary point P(x, y, z), the position vector measured from the origin O to P is r = xi + yj + zk.
• Any two vectors boasting equivalent magnitude also share the same direction; they are considered equal.
• Given vectors P =x1i + x2j + x3k and Q = y1i + y2j + y3k, addition and subtraction are performed on corresponding components.
• Sum: P + Q = (x1 + y1)i + (x2 + y2)j + (x3 + y3)k.
• The triangle law posits that with any origin O, vectors a and b represented by OA and OB, their sum a+b is given by OB = OA + AB.
• Taking 2 vectors from a triangle in order yields a third sum, taken in opposite directions.
• To subtract vector b from a, the difference a-b means summing a and -b.
• The sum of the two vectors a and b, in the figure, occurs by placing the terminal point of a on the initial point of b and then joining the initial point of a to the terminal point of b.

Types of vectors

• The sum of multiple vectors, such as p, q, r and s, is produced by connecting each vector terminal point to the next vector's initial point.
• A position vector of point P relative to an origin O uniquely defines P’s position in reference to O and is denoted by OP.
• The null vector's modulus equals zero and is also known as the zero vector.

Parallel vectors

• Given non-null vectors A and B, A = pB means that A and B are parallel.
• "p" in p = +/- [A/B] denotes the same sense for positive values and opposite sense if negative.

Scalar Multiplication

• Given a scalar m, the scalar multiple vector mp possesses a magnitude scaled by m.
• The direction depends on m’s sign; negative m reverses direction, and m=0 results in a null vector.
• Given vector space R" and vector u = (u1, u2...un), product mu = (mu1, mu2...mum).

Vector Laws

• For vectors p, q, r and scalars x, y associative, commutative, distributive laws apply:

• (p+q) + r = p + (q+r) holds

• p + O = O + p = p (zero vector) holds

• p + (-p) = (-p) + p = 0 holds

• x (p + q) = xp + xq holds

• (x + y)p = xp + yp holds

• m (np) = (mn)p holds

• unity I (A) = A (A = unity) holds

• If p+r = q+r, then p = q

Linear Combination of Vectors

• Linear combinations involve summing scalar products of vectors.
• A linear combination is described as Q = m1P1 + m2P2 + ... + mnPn, combining vectors P1, P2, ... , Pn with scalars.
• The vectors become linearly dependent if scalars m1, m2 ..., mn exist (not all zeroes) such that m1P1 + m2P2 + ... + mnPn = 0.
• Conversely, if all scalars must be zero for the sum to vanish, the vectors are linearly independent.

Coplanar Vectors

• Coplanar vectors are parallel, residing on one plane.
• A coplanar vector r equals xa + yb, given non-collinear vectors a and b, where x and y are respective scalars.
• In R", given vector u= (u1, u2 … un) and scalar m, scalar multiple operation mu= (mu1, mu2 … mun).
• If three vectors reside on the same plane, then they are coplanar and can express some relationship from one to another.

Scalar multiplication

• Modulus |A| = √(3^2 + (−2)^2 + (3)^2) yields magnitude of vector A
• Scalar multiplication gives the size given the scale, as well as the length and direction of the new scalar.

Direction Cosines

• Direction cosines involve cosines of angles made by a vector q with coordinate axes:

• Given q = xi + yj + zk and direction angles α, β,and γ, then the magnitude |q| equals √x^2 + y^2 + z^2.

• Then cos α = x/|q|, cos β = y/|q|, and cos γ = z/|q| are direction cosines.

Straight Line equation

• The position vector of any line point C, where line is drawn through points A and B, gives the equation r = (1-t) a = tb. The plane's partition has the line equation ax + by + c = 0.

Equation of a Straight Line

• Given two points A, B and position vector r of any point C on the straight line: vector equation r=(1 – t)a + tb. Scalars m and n yield m(r – a) = n(b – r), and scalars can express the result in symmetric form.
• Vector r, relative to points A and B with originating positions a and b, satisfies: m(AC)=n(CB).

Chapter 2: Scalar and Vector Products

• Scalar and vector (cross) products represent two ways to multiply vectors p and q. Dot operation, dot product, calculates 3 components, producing a real number. For example, in scalar format p-q is measured "p dot q”. Product equation: p.q = |p| |q| cos θ = pq cos θ.
• Dot law: p.q = ±pq for parallel vectors and p.q = 0 perpendicular to direction components. theta = 0 and theta = pi, in directions.
• In dot function, P = Xi + Yj Q = Xi+ Yj vector multiplication and component direction. p.q = (Xi + Yj )(Xi + Yj)
• Vectors and scalars can form direction, magnitude, and angular properties and must remain scalable under dot product.

Dot Product

• In scalar format equation, p.q = sqrt2(x-x1)
• Determine value x of equation p= xi +2j, to equate scalar of dot product, component addition and 90.

Chapter 3: Differentiation and Integration

• Vector functions show derivative functions with standard law of differentiation for each scalar to derivative function. If U, V, W = vector d/dt and theta = scalar functions. a =scalar and product value = d/dt A v = AV d/dt, shows component value of d and v. Derivative can be expressed as (Ax) = 0 is constant. d /dt (U v) is derivative scalar operation by du/dt etc... equation 1-7.

Linear Velocities

• Linear velocity involves differentiation of a vector describing displacement in terms of vector functions.
• A vector is linear when direction at a point contains time expression and initial acceleration.
• To determine force and angular velocity: wxr (vector), a= angular velocity. Vectors of direction and magnitude can be observed with each term and combined.

Moment of Force

• PQ = vector (line of action); in plane containing Q about point A.

Integration of vector functions

• Derivatives of the integration of vector value functions. Vector and scalar terms integrated and components given.
• If r * d /dt(time) + ½ r*r (vector value function equation), set between two points then values determine components by setting = 0.

Equation

• Equation must then be solved using basic methods.
• To determine a line, equate the number of sides and value of x in scalar to scalar values.
• Unit vector perpendicular uses vector functions and derivatives and must then show equations to equal non coplanar.