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Vectors and Vector Spaces

Subject: Mathematics
Topic: 8
Cambridge Code: 0580

Vector Fundamentals​

Vector - Quantity with magnitude and direction

Notation​

Column vector: v=(abc)\mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}

Component form: v=ai+bj+ck\mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}

Directional form: AB→\overrightarrow{AB} (from A to B)

Position Vectors​

Position vector with origin O: OA→=(a1a2a3)\overrightarrow{OA} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}

Vector from Point to Point​

AB→=B−A=(b1−a1b2−a2b3−a3)\overrightarrow{AB} = B - A = \begin{pmatrix} b_1 - a_1 \\ b_2 - a_2 \\ b_3 - a_3 \end{pmatrix}

Vector Operations​

Addition and Subtraction​

u+v=(u1+v1u2+v2u3+v3)\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix}

Graphically: Tip-to-tail method

Scalar Multiplication​

kv=(kv1kv2kv3)k\mathbf{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix}

Properties:

  • k > 0: Same direction, scaled magnitude
  • k < 0: Opposite direction
  • |k|: Magnitude factor

Magnitude (Length)​

Magnitude Formula​

∣v∣=v12+v22+v32|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}

2D: ∣v∣=v12+v22|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}

Unit Vectors​

Unit vector - Magnitude 1

v^=v∣v∣\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}

∣v^∣=1|\hat{\mathbf{v}}| = 1

Standard Basis Vectors​

i=(100),j=(010),k=(001)\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}

Dot Product (Scalar Product)​

Dot product - Scalar result from two vectors

Definition​

u⋅v=u1v1+u2v2+u3v3\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3

Or using angle: u⋅v=∣u∣∣v∣cos⁡θ\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos θ

Angle Between Vectors​

cos⁡θ=u⋅v∣u∣∣v∣\cos θ = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}

Orthogonality​

Orthogonal vectors - Perpendicular (θ=90°θ = 90°)

u⋅v=0 iff u⊥v\mathbf{u} \cdot \mathbf{v} = 0 \text{ iff } \mathbf{u} ⊥ \mathbf{v}

Properties​

  • Commutative: u⋅v=v⋅u\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}
  • Distributive: u⋅(v+w)=u⋅v+u⋅w\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}
  • Scalar multiplication: (ku)⋅v=k(u⋅v)(k\mathbf{u}) \cdot \mathbf{v} = k(\mathbf{u} \cdot \mathbf{v})

Cross Product (Vector Product)​

Cross product - Vector result from two 3D vectors

Definition​

u×v=∣ijku1u2u3v1v2v3∣\mathbf{u} × \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}

=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k= (u_2v_3 - u_3v_2)\mathbf{i} - (u_1v_3 - u_3v_1)\mathbf{j} + (u_1v_2 - u_2v_1)\mathbf{k}

Magnitude and Angle​

∣u×v∣=∣u∣∣v∣sin⁡θ|\mathbf{u} × \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin θ

Direction given by right-hand rule

Properties​

  • Anti-commutative: u×v=−(v×u)\mathbf{u} × \mathbf{v} = -(\mathbf{v} × \mathbf{u})
  • Distributive: u×(v+w)=u×v+u×w\mathbf{u} × (\mathbf{v} + \mathbf{w}) = \mathbf{u} × \mathbf{v} + \mathbf{u} × \mathbf{w}
  • Parallel vectors: u×v=0\mathbf{u} × \mathbf{v} = \mathbf{0} if parallel

Geometric Interpretation​

∣u×v∣|\mathbf{u} × \mathbf{v}| = Area of parallelogram formed by u and v

Lines in 3D​

Parametric Equation​

r(t)=a+td\mathbf{r}(t) = \mathbf{a} + t\mathbf{d}

Where:

  • a = point on line
  • d = direction vector
  • t = parameter

Cartesian Equation​

If direction = (a, b, c) and passes through (x0,y0,z0)(x_0, y_0, z_0):

x−x0a=y−y0b=z−z0c\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}

Angle Between Lines​

cos⁡θ=∣d1⋅d2∣∣d1∣∣d2∣\cos θ = \frac{|\mathbf{d_1} \cdot \mathbf{d_2}|}{|\mathbf{d_1}||\mathbf{d_2}|}

Planes in 3D​

Equation of Plane​

Normal form: n⋅(r−a)=0\mathbf{n} \cdot (\mathbf{r} - \mathbf{a}) = 0

Standard form: ax+by+cz=dax + by + cz = d

Where (a, b, c) is normal vector

Angle Between Planes​

Equal to angle between normals: cos⁡θ=∣n1⋅n2∣∣n1∣∣n2∣\cos θ = \frac{|\mathbf{n_1} \cdot \mathbf{n_2}|}{|\mathbf{n_1}||\mathbf{n_2}|}

Vector Spaces​

Vector space - Set closed under addition and scalar multiplication

Properties​

For vectors u, v, w and scalars a, b:

  1. Closure under addition
  2. Commutativity: u + v = v + u
  3. Associativity: (u + v) + w = u + (v + w)
  4. Zero vector exists
  5. Closure under scalar multiplication
  6. Distributivity: a(u + v) = au + av

Subspaces​

Subspace - Subset that is also vector space

Must contain zero vector and closed under operations

Basis and Dimension​

Basis - Set of linearly independent vectors spanning space

Dimension - Number of basis vectors

Example: ℝ³ has dimension 3

Linear Independence​

Linearly independent - No vector is linear combination of others

Linearly dependent - One vector is combination of others

Testing Independence​

Vectors v₁, v₂, ..., vₙ independent iff: c1v1+c2v2+...+cnvn=0c_1\mathbf{v_1} + c_2\mathbf{v_2} + ... + c_n\mathbf{v_n} = \mathbf{0}

Only when all ci=0c_i = 0

Applications​

Physics​

Velocity, acceleration, force vectors

Engineering​

Stress, strain, and structural analysis

Graphics​

Vector operations in 3D rendering

Direction and distance calculations

Key Points​

  1. Vector has magnitude and direction
  2. Addition: Tip-to-tail or component-wise
  3. Dot product gives scalar
  4. Dot product zero means perpendicular
  5. Cross product gives perpendicular vector
  6. Parametric equations define lines
  7. Planes have normal vectors
  8. Vector spaces satisfy closure properties
  9. Basis vectors are linearly independent
  10. Dimension = number of basis vectors

Practice Questions​

  1. Calculate magnitudes
  2. Find unit vectors
  3. Compute dot products
  4. Calculate cross products
  5. Find angles between vectors
  6. Write line equations
  7. Write plane equations
  8. Check linear independence
  9. Find bases for spaces
  10. Apply to physical problems

Revision Tips​

  • Visualize in 3D when possible
  • Remember right-hand rule for cross product
  • Know when to use dot vs cross product
  • Practice component calculations
  • Understand geometric interpretations
  • Work with parametric equations
  • Test perpendicularity with dot product
  • Verify answers reasonably
  • Connect to applications