Vector notation, components, and basic operations

Key idea

A vector in \(\mathbb{R}^n\) can be written as \(v=(v_1,v_2,\dots,v_n)\). The core operations are component-wise: \[ (u_1,\dots,u_n) + (v_1,\dots,v_n) = (u_1+v_1,\dots,u_n+v_n), \] \[ (u_1,\dots,u_n) - (v_1,\dots,v_n) = (u_1-v_1,\dots,u_n-v_n), \] and for a scalar \(k\), \[ k(v_1,\dots,v_n) = (kv_1,\dots,kv_n). \]

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Summary

• Add/subtract vectors by adding/subtracting corresponding components.
• Scalar multiplication multiplies every component by the scalar.

Magnitude (length) and unit vectors

Key idea

The magnitude (Euclidean norm) of \(v=(v_1,v_2,\dots,v_n)\) is: \[ \|v\|=\sqrt{v_1^2+v_2^2+\cdots+v_n^2}. \] For a nonzero vector \(v\), the unit vector in the direction of \(v\) is: \[ \hat v = \frac{v}{\|v\|}. \] This keeps direction the same but rescales length to \(1\).

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Summary

• Magnitude: \(\|v\|=\sqrt{\sum v_i^2}\).
• Unit vector: \(\hat v=\dfrac{v}{\|v\|}\) for v≠ 0.

Dot product, angles, and orthogonality

Key idea

The dot product of \(u=(u_1,\dots,u_n)\) and \(v=(v_1,\dots,v_n)\) is: \[ u\cdot v=u_1v_1+\cdots+u_nv_n. \] It connects algebra to geometry: \[ u\cdot v=\|u\|\,\|v\|\cos\theta, \] where \(\theta\) is the angle between \(u\) and \(v\) (for nonzero vectors). In particular, orthogonal vectors satisfy \(u\cdot v=0\).

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Summary

• Dot product: \(u\cdot v=\sum u_iv_i\).
• Orthogonal: \(u\cdot v=0\). Angle: \(\cos\theta=\dfrac{u\cdot v}{\|u\|\|v\|}\).

Vector projection and components along a direction

Key idea

The projection of \(a\) onto a nonzero vector \(b\) is: \[ \mathrm{proj}_b a = \frac{a\cdot b}{b\cdot b}\,b. \] This gives the vector component of \(a\) that points in the direction of \(b\). (If you only want the signed length, the scalar projection is \(\mathrm{comp}_b a = \dfrac{a\cdot b}{\|b\|}\).)

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Summary

• Projection formula: \(\mathrm{proj}_b a=\dfrac{a\cdot b}{b\cdot b}\,b\) for b≠ 0.
• Projection isolates the part of \(a\) that points along \(b\).

Orthonormal sets and linear independence

Key idea

A set of vectors is orthonormal if every vector has length \(1\) and every pair is orthogonal: \[ \|u\|=\|v\|=1 \quad \text{and} \quad u\cdot v=0. \] Vectors are linearly dependent if one is a scalar multiple of another (for example, \(v=2u\)). Two vectors in \(\mathbb{R}^2\) are linearly independent precisely when they are not multiples.

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Summary

• Orthonormal: each vector has length \(1\) and every pair has dot product \(0\).
• Linear dependence: one vector is a scalar multiple of another (common quick test).

Cross product basics in \(\mathbb{R}^3\)

Key idea

The cross product \(u\times v\) is defined for vectors in \(\mathbb{R}^3\). It produces a vector perpendicular to both \(u\) and \(v\). Its magnitude is: \[ \|u\times v\|=\|u\|\,\|v\|\sin\theta, \] which equals the area of the parallelogram spanned by \(u\) and \(v\). The direction is determined by the right-hand rule.

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Summary

• \(u\times v\) is perpendicular to both \(u\) and \(v\) (in \(\mathbb{R}^3\)).
• \(\|u\times v\|=\|u\|\|v\|\sin\theta\) gives area of a parallelogram.

Why vector operations matter

Where vectors show up

• Geometry: displacement, direction, distance, and angles.
• Physics: forces, velocity, acceleration, and component resolution using projections.
• Computer graphics: lighting and viewing angles via dot products; normals via cross products.
• Data and ML: cosine similarity (a normalized dot product) for comparing directions in high-dimensional spaces.

Worked example: angle from orthogonality

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Final recap

• Vectors are added/subtracted component-by-component; scalars multiply every component.
• Magnitude: \(\|v\|=\sqrt{\sum v_i^2}\); unit vector: \(\hat v=\dfrac{v}{\|v\|}\) for v≠ 0.
• Dot product connects algebra to geometry: \(u\cdot v=\|u\|\|v\|\cos\theta\) and orthogonal means \(u\cdot v=0\).
• Projection: \(\mathrm{proj}_b a=\dfrac{a\cdot b}{b\cdot b}\,b\) for b≠ 0.
• Cross product in \(\mathbb{R}^3\) gives a perpendicular vector and area magnitude.