A vector has magnitude, direction, and obeys laws of vector algebra. Unit vector â = A/|A| has magnitude 1. Zero vector has zero magnitude and arbitrary direction.

• Current has magnitude and direction but is a scalar because it does not obey the parallelogram law of vector addition — a standard NEET assertion-reason trap.
• Polar vectors (displacement, force) have a point of application; axial vectors (torque, angular momentum) lie along the rotation axis per the right-hand rule.
• NEET trap: confusing collinear vectors (angle 0° or 180° between them) with coplanar vectors (lying in the same plane) — all collinear vectors are coplanar, but not vice versa.

R = √(A² + B² + 2AB cos θ), direction: tan α = B sin θ/(A + B cos θ), where θ is the angle between vectors A and B.

• The two vectors must be placed tail-to-head (same order) for the triangle law to apply — the resultant is the closing side drawn from the tail of the first to the head of the second.
• When θ = 90°, the formula reduces to the Pythagorean form R = √(A²+B²), which is the fastest shortcut in NEET.
• NEET trap: using the wrong angle — θ in the formula is the angle between the vectors when placed tail-to-tail, not the interior angle of the triangle.

Two vectors as adjacent sides of a parallelogram → resultant is the diagonal. Same magnitude formula: R = √(A² + B² + 2AB cos θ). Special cases: θ=0° → R=A+B; θ=180° → R=|A−B|; θ=90° → R=√(A²+B²).

• The parallelogram law and triangle law give identical magnitude and direction — the geometric construction differs but the algebra is the same.
• Maximum resultant occurs when vectors are parallel (θ=0°); minimum when anti-parallel (θ=180°).
• NEET trap: forgetting that R_min = |A−B|, not A−B, and that R lies between |A−B| and A+B for any angle θ.

If (n−1) vectors form (n−1) sides of an n-sided polygon in order, the resultant is the nth (closing) side in the opposite order. R = A + B + C + D + …

• The polygon law is a generalisation of the triangle law applied repeatedly — each successive vector is placed head-to-tail.
• If n vectors form a closed polygon (all n sides in order), their resultant is zero — this is the equilibrium condition for concurrent forces.
• NEET trap: concluding that three non-coplanar vectors can sum to zero — they cannot, because a polygon of three sides must be planar.

A − B = A + (−B). Magnitude: |A − B| = √(A² + B² − 2AB cos θ). Direction: tan α₂ = B sin θ/(A − B cos θ).

• Vector subtraction reverses B and then adds — the magnitude formula replaces +2AB cos θ with −2AB cos θ.
• If |A+B| = |A−B|, then cos θ = 0, so θ = 90° — vectors are perpendicular. NEET uses this as a quick conceptual MCQ.
• NEET trap: assuming |A−B| = A−B. This holds only when A and B are parallel (θ=0°); otherwise the magnitude formula must be used.

R = R_x î + R_y ĵ where R_x = R cos θ, R_y = R sin θ. Inverse: R = √(R_x²+R_y²), θ = tan⁻¹(R_y/R_x).

• The angle θ is always measured from the axis along which the cosine component is taken — if measured from the y-axis, R_x = R sin θ and R_y = R cos θ.
• On an inclined plane, resolve gravity along and perpendicular to the plane: mg sin α (along) and mg cos α (perpendicular) — this is the most-tested NEET application.
• NEET trap: swapping sin and cos when the reference axis changes from x to y or from horizontal to the incline.

R = R_x î + R_y ĵ + R_z k̂. Direction cosines: l = cos α = R_x/R, m = cos β = R_y/R, n = cos γ = R_z/R. Constraint: l² + m² + n² = 1.

• The constraint cos²α + cos²β + cos²γ = 1 serves as a quick verification — if your direction cosines do not satisfy it, recheck the component calculation.
• Position vector of point (x,y,z) is r = xî + yĵ + zk̂; displacement vector from (x₁,y₁,z₁) to (x₂,y₂,z₂) is Δr = (x₂−x₁)î + (y₂−y₁)ĵ + (z₂−z₁)k̂.
• NEET trap: computing the magnitude as R_x + R_y + R_z instead of √(R_x²+R_y²+R_z²) — vector magnitudes require the Pythagorean sum, not arithmetic addition.

A·B = AB cos θ = A_xB_x + A_yB_y + A_zB_z. Properties: commutative (A·B = B·A), î·î = 1, î·ĵ = 0. Applications: W = F·s, P = F·v.

• If A·B = 0 and neither A nor B is zero, then θ = 90° — the vectors are perpendicular. This is the orthogonality test NEET frequently uses.
• The component form A_xB_x + A_yB_y + A_zB_z is faster than AB cos θ when vectors are given in î, ĵ, k̂ notation.
• NEET trap: computing work as |F|×|s| without cos θ — work uses the dot product, so W = Fs cos θ, not Fs.

A×B = AB sin θ n̂. Determinant: A×B = |î ĵ k̂; A_x A_y A_z; B_x B_y B_z|. Non-commutative: A×B = −B×A. Self product: A×A = 0.

• The direction of A×B is given by the right-hand screw rule: curl fingers from A to B through the smaller angle, and the thumb points along A×B.
• |A×B| gives the area of the parallelogram formed by A and B; ½|A×B| gives the area of the triangle.
• NEET trap: ignoring the non-commutative property — A×B = −(B×A), so reversing the order flips the direction. This matters in torque (r×F ≠ F×r) and magnetic force (qv×B ≠ qB×v).

If A + B + C = 0 (equilibrium), then A/sin α = B/sin β = C/sin γ, where α, β, γ are angles opposite to vectors A, B, C respectively.

• Lami's theorem applies only to exactly three concurrent coplanar forces in equilibrium — verify all three conditions before applying.
• The angles α, β, γ are the angles between the other two forces (opposite angles), not the angles each force makes with a reference axis.
• NEET trap: confusing the angle between two forces with the angle opposite to a force — if forces B and C make angle α between them, then α is the angle opposite to force A in the Lami's construction.