Vectors

Precise definition

A vector in $\mathbb R^n$ is an ordered tuple with componentwise addition and scalar multiplication. Geometrically it may represent displacement, velocity, or force, independent of its drawing location. The Euclidean norm is $\|v\|=\sqrt{v_1^2+\cdots+v_n^2}$, and a unit vector has norm 1.

Notation and mathematical language

The dot product is $u\cdot v=\sum u_iv_i=\|u\|\|v\|\cos\theta$. It measures alignment and defines orthogonality. A linear combination $au+bv$ combines directions with scalar weights. Components depend on the chosen coordinate axes and units.

Conceptual picture

Vector addition follows head-to-tail displacement; scalar multiplication changes magnitude and may reverse direction. The dot product converts two vectors to a scalar and connects algebraic components with angle and projection.

Conditions and key results

The cosine formula for angle requires nonzero vectors and Euclidean inner product. Physical vector addition requires compatible units. Points and position vectors are related but conceptually distinct: subtracting points produces a displacement vector.

A reliable strategy

1. Choose coordinates and state what each component and unit represents.
2. Perform componentwise addition or scaling; compute norm or dot product from definitions.
3. For angles, divide by both nonzero norms and control calculator mode.
4. Check geometry, units, signs, and reconstruct the intended displacement or resultant.

Fully worked example

Interpretation and application

Vectors model forces, navigation, data points, and states. A component can change under rotation of axes while the physical vector and its norm remain unchanged. Context determines whether Euclidean distance is meaningful for data.

Common mistakes

Verification and reasonableness

• Use the triangle inequality $\|u+v\|\le\|u\|+\|v\|$ as a magnitude check.
• Recompute a dot product from geometry when an angle is known.
• Check units and verify a normalized vector has norm 1.

Practice

1. Find the norm of $(3,4)$.
2. What is $(1,2)+(4,-1)$?
3. When are nonzero vectors orthogonal?

Further deduction

Cauchy–Schwarz, $|u\cdot v|\le\|u\|\|v\|$, guarantees the angle cosine lies between $-1$ and 1. Equality holds exactly when the vectors are linearly dependent. The triangle inequality follows from it, showing how an algebraic inner-product bound controls Euclidean geometry.

The cross product in $\mathbb R^3$ produces a vector perpendicular to two inputs with magnitude equal to parallelogram area and orientation from the right-hand rule. It is not the general multiplication operation for vectors in every dimension. The dot product exists in any finite Euclidean dimension, while the familiar cross product is dimension-specific.

Points and displacement vectors have different roles even when both are stored as coordinate lists. Subtracting points $Q-P$ produces a displacement, and adding that displacement back to $P$ produces $Q$; adding two points has no coordinate-independent geometric meaning. Under a change of origin, point coordinates change while a displacement's components do not. This affine distinction matters in geometry, graphics, and mechanics and prevents treating every ordered tuple as the same kind of object.

Orientation can be encoded by the sign of a determinant: in the plane, $\det[u\ v]>0$ means the ordered turn from $u$ to $v$ is counterclockwise. Swapping the vectors reverses the sign while preserving parallelogram area magnitude.

Related topics

• Span
• Orthogonality
• Vector Spaces