Vector Quantities

A vector quantity is a physical quantity characterised by two inseparable pieces of information: (1) its magnitude — a non-negative number together with a physical unit — and (2) its direction in space. Neither piece alone is sufficient. A force of 10 N is incomplete without knowing which way it pushes; a direction 'upward' is meaningless without a size. Both must be specified simultaneously for the quantity to be physically meaningful.

Several equivalent notations exist. In printed text, vectors appear in bold: F, v, a. In handwritten work, an arrow is drawn above the symbol: F⃗, v⃗, a⃗. The magnitude of a vector v is written as |v| or simply v (in italic, without bold or arrow). You must never confuse v (magnitude, a scalar) with v (the full vector). When a vector is given in component form in two dimensions, it is written as v = (vₓ, v_y) or using unit-vector notation as v = vₓ î + v_y ĵ, where î and ĵ are unit vectors pointing in the positive x- and y-directions respectively.

Geometrically, a vector is represented by a directed line segment — an arrow. The length of the arrow (drawn to a chosen scale) represents the magnitude, and the arrowhead shows the direction. The tail of the arrow is the starting point and the tip (head) is the endpoint. For instance, a displacement vector of 5 m east would be drawn as an arrow 5 cm long (at a scale of 1 cm : 1 m) pointing to the right. Crucially, a vector is defined only by its magnitude and direction, NOT by its location in space — you may slide an arrow anywhere on a diagram without changing the vector it represents, as long as you preserve its length and direction.

The most frequently encountered vector quantities in high school physics are: displacement (Δr, measured in metres), velocity (v, in m/s), acceleration (a, in m/s²), force (F, in newtons N), momentum (p, in kg·m/s), and gravitational field strength (g, in N/kg or m/s²). Each of these requires a direction to be physically complete. Compare these with scalar counterparts: distance (m), speed (m/s), mass (kg), and kinetic energy (J) — none of which have an associated direction.

Because vectors carry directional information, they do not obey ordinary algebraic rules. Two vectors can only be added using the triangle rule (tip-to-tail method) or parallelogram rule, or by resolving them into components and adding components separately. For example, a displacement of 3 m east followed by 4 m north does not give a total displacement of 7 m — it gives a displacement of 5 m in a direction 53.1° north of east (by the Pythagorean theorem: √(3² + 4²) = 5 m). This is a defining feature that separates vector algebra from scalar arithmetic.

Multiplying a vector by a positive scalar changes its magnitude but not its direction. Multiplying by a negative scalar reverses its direction. The vector −v has the same magnitude as v but points in exactly the opposite direction. For example, if v = 10 m/s north, then −v = 10 m/s south. Multiplying v by the scalar 2 gives 2v = 20 m/s north. These operations are essential for understanding vector subtraction (which is defined as adding the negative vector) and for decomposing vectors into components.

• \vec{v} — Vector quantity (depends on context (e.g., m/s for velocity)): The full vector, specifying both magnitude and direction.
• v_x — x-component (same as the vector's unit): The signed scalar projection of the vector onto the positive x-axis. Can be positive, negative, or zero.
• v_y — y-component (same as the vector's unit): The signed scalar projection of the vector onto the positive y-axis. Can be positive, negative, or zero.
• \hat{\mathbf{i}} — Unit vector in x-direction (dimensionless): A vector of magnitude exactly 1 with no units, pointing in the positive x-direction. It carries directional information only.
• \hat{\mathbf{j}} — Unit vector in y-direction (dimensionless): A vector of magnitude exactly 1 with no units, pointing in the positive y-direction.

Problem. A drone flies 6.0 m due east, then 8.0 m due north. (a) Represent each displacement as a vector in component form using î and ĵ notation. (b) Find the resultant displacement vector. (c) Calculate the magnitude and direction (angle above east) of the resultant displacement.

Answer. Resultant displacement = 10.0 m at 53.1° north of east

• Scalar Quantities — Scalars are the complementary category to vectors — they have magnitude only, no direction. Understanding both categories together clarifies exactly what additional information a vector carries and why special rules are needed to manipulate it.
• Position Vector — The position vector is one of the most fundamental vector quantities, specifying the location of a point relative to an origin using both magnitude and direction. It is the first direct application of vector notation in kinematics.
• Displacement Vector — Displacement is a vector quantity that records the change in position, including direction. It is the primary example used to introduce vector ideas in kinematics and distinguishes vector quantities from their scalar counterparts like distance.
• Vector Equality & Scaling — Two vectors are equal only if they share both the same magnitude and the same direction. Scaling multiplies magnitude while preserving or reversing direction. These ideas deepen understanding of what a vector fundamentally is.
• Graphical Vector Addition — Once you know what vectors are, the next step is combining them. Graphical addition (tip-to-tail and parallelogram methods) directly uses the arrow representation of vectors to find resultants, reinforcing the directional nature of vector quantities.
• Analytical Vector Addition — Analytical addition uses component form — the algebraic expression of vectors — to calculate resultants precisely. This requires a firm understanding of vector notation introduced in this concept.
• Resolution of Vectors — Resolution is the process of splitting a single vector into its components along chosen axes. It relies directly on the definition of a vector and is the reverse of vector addition, making it an essential technique for problem-solving.
• Distance vs Displacement — This concept uses the scalar–vector distinction in a concrete kinematic context. Distance is the scalar total path length; displacement is the vector change in position. Comparing them shows why the vector nature of displacement matters.
• Speed vs Velocity — Speed is the scalar magnitude of motion; velocity is the vector quantity that includes direction of motion. This pair is one of the most important illustrations of the scalar-versus-vector distinction in kinematics.
• Average Acceleration — Acceleration is defined as the rate of change of the velocity vector, making it inherently a vector quantity. Understanding that acceleration has direction — not just magnitude — requires a solid grasp of what vector quantities are.