Scalars and Vectors

Scalar vs vector — the operational definition

A scalar has magnitude only (mass, time, energy, temperature); a vector has magnitude and direction and obeys the parallelogram law of addition (displacement, velocity, force, momentum). The Punjab textbook Chapter 2 and HRW Chapter 3 both stress that the test is not whether a quantity has a direction in some loose sense, but whether it adds vectorially. Electric current has a direction along a wire yet is a scalar because two currents at a junction add algebraically, not by the parallelogram law.

Resolution and components

Any vector A making angle θ with the x-axis resolves into A_x = A cos θ and A_y = A sin θ. The reverse: |A| = √(A_x² + A_y²) and tan θ = A_y/A_x. A 50 N force at 37° above the horizontal has components 50·cos 37° ≈ 40 N horizontal and 50·sin 37° = 30 N vertical (the 3-4-5 triangle is a perennial MDCAT favourite). When several forces act, sum components first, then recombine — never average angles directly.

Unit vectors and rectangular form

Writing A = A_xî + A_yĵ + A_zk̂ converts vector arithmetic into algebra. Addition is component-wise: (3î + 4ĵ) + (1î − 2ĵ) = 4î + 2ĵ, magnitude √20 ≈ 4.47. The unit vector along A is  = A/|A|; for 3î + 4ĵ this is (3î + 4ĵ)/5 = 0.6î + 0.8ĵ. MDCAT MCQs often hide a check: the magnitude of any unit vector must be exactly 1.

Dot product and cross product

Scalar (dot) product: A·B = AB cos θ = A_xB_x + A_yB_y + A_zB_z. It is commutative, distributive, and zero for perpendicular vectors. Work W = F·d uses it. Vector (cross) product: A×B = AB sin θ n̂, with n̂ given by the right-hand rule. It is anti-commutative (A×B = −B×A), zero for parallel vectors, and maximal for perpendicular ones. Torque τ = r×F and angular momentum L = r×p are cross products. î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ — cycle forward positive, backward negative.

Worked numerical patterns

Two forces 3 N east and 4 N north have resultant √(9+16) = 5 N at tan⁻¹(4/3) ≈ 53° north of east. Three vectors of equal magnitude 120° apart (like the symmetry of an equilateral triangle) sum to zero — this is the basis of the Lami's-theorem MCQs that appear roughly every other year. The angle between A = 2î + 3ĵ and B = î − ĵ is found from cos θ = (A·B)/(|A||B|) = (2−3)/(√13·√2) = −1/√26, so θ ≈ 101°. Negative dot product means the angle is obtuse — a frequent trap.

Common traps you must internalise

The magnitude of a sum is not the sum of magnitudes unless the vectors are parallel; for anti-parallel vectors it is the difference. The maximum and minimum resultants of two vectors of magnitudes a and b are a+b and |a−b| respectively. A vector with all three components equal in magnitude makes an angle cos⁻¹(1/√3) ≈ 54.7° with each axis, not 60°. Cross product of a vector with itself is the zero vector, not zero scalar. Always cite Serway & Jewett Physics for Scientists and Engineers or HRW for the formal proofs; the FSc Punjab Textbook gives the operational shortcuts MDCAT actually tests.