Q.Find the scalar and vector products of two vectors a = (3i - 4j + 5k) and

A scalar is fully specified by a magnitude (mass, speed, distance, work, energy, temperature, charge, current — note current is a scalar despite having a direction of flow). A vector needs a magnitude and a direction and obeys the triangle law of addition (displacement, velocity, acceleration, force, momentum, torque). Vector algebra is the toolkit for two-dimensional kinematics, and JEE Main tests every piece of it.

1 — Addition: the resultant. Two vectors A and B at an angle θ have a resultant of magnitude

R = √(A² + B² + 2AB·cosθ), directed at tan α = B·sinθ / (A + B·cosθ) from A.

Special cases worth memorising: θ = 0 → R = A + B (maximum); θ = 180° → R = |A − B| (minimum); θ = 90° → R = √(A²+B²); two equal vectors of magnitude A give R = 2A·cos(θ/2), so at 120° they give exactly A. The resultant of two vectors can be any value between |A − B| and A + B.

2 — Subtraction. A − B = A + (−B), so |A − B| = √(A² + B² − 2AB·cosθ). This is how you get a change in a vector: the change in velocity Δv = v_f − v_i (e.g. a ball reversing direction has |Δv| = 2v). Two useful conditions: |A + B| = |A − B| ⇔ A ⊥ B; |A + B| = |A| + |B| ⇔ A ∥ B.

3 — Resolution and components. A vector A at angle θ to the x-axis has components Aₓ = A·cosθ, A_y = A·sinθ; conversely A = √(Aₓ² + A_y²) and θ = tan⁻¹(A_y/Aₓ). In three dimensions A = Aₓî + A_yĵ + A_zk̂, magnitude √(Aₓ²+A_y²+A_z²), and the unit vector  = A/|A|. The direction cosines cosα, cosβ, cosγ (with the three axes) satisfy cos²α + cos²β + cos²γ = 1; a vector equally inclined to all three axes has cosα = 1/√3.

4 — The scalar (dot) product. A·B = AB·cosθ = aₓbₓ + a_yb_y + a_zb_z. It is commutative, gives the angle between two vectors (cosθ = A·B/|A||B|), the projection of A along B (A·B/|B|), and physical scalars like work W = F·d. Two vectors are perpendicular ⇔ A·B = 0. …