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Vectors and matrices Knowledge-Based Study

Math · Functions & Trigonometry · Math · Polar, Parametric & Vectors

Math topic explanations on Vectors and matrices.

Study notes

  1. Vectors in the Plane

    A vector represents a quantity with both magnitude and direction. In the Cartesian coordinate plane, a two-dimensional vector v⃗\vec{v} with initial point at the origin (0,0)(0,0) and terminal point at (x,y)(x, y) is written in component form as:

    v⃗=(xy)=⟨x,y⟩\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix} = \langle x, y \rangle

    The magnitude (or length) of v⃗\vec{v}, denoted ∥v⃗∥\|\vec{v}\|, is computed using the distance formula:

    ∥v⃗∥=x2+y2\|\vec{v}\| = \sqrt{x^2 + y^2}

    Its direction angle θ\theta, measured counterclockwise from the positive xx-axis, satisfies:

    tan⁡(θ)=yx(x≠0)\tan(\theta) = \frac{y}{x} \quad (x \neq 0)

    A unit vector in the direction of v⃗\vec{v} is given by u⃗=1∥v⃗∥v⃗\vec{u} = \frac{1}{\|\vec{v}\|}\vec{v}.

    Vector Arithmetic and Inner Product

    Given vectors u⃗=(u1u2)\vec{u} = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} and v⃗=(v1v2)\vec{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}, along with a scalar c∈Rc \in \mathbb{R}:

    • Vector Addition: u⃗+v⃗=(u1+v1u2+v2)\vec{u} + \vec{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \end{pmatrix}
    • Scalar Multiplication: cu⃗=(cu1cu2)c\vec{u} = \begin{pmatrix} c u_1 \\ c u_2 \end{pmatrix}

    The dot product (or scalar product) is defined algebraically as:

    u⃗⋅v⃗=u1v1+u2v2\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2

    Geometrically, it relates to the angle θ\theta between the two vectors:

    u⃗⋅v⃗=∥u⃗∥∥v⃗∥cos⁡(θ)\vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos(\theta)

    Two nonzero vectors are orthogonal (perpendicular) if and only if u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0.

    Matrix-Vector Multiplication

    A 2×22 \times 2 matrix can act on a two-dimensional vector as a linear transformation mapping the plane onto itself:

    A=(abcd),v⃗=(xy)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad \vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}

    The product Av⃗A\vec{v} is evaluated by taking linear combinations of the columns of AA:

    Av⃗=(abcd)(xy)=(ax+bycx+dy)A\vec{v} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}

    Standard transformations in the plane include:

    • Counterclockwise Rotation by θ\theta: Rθ=(cos⁡(θ)−sin⁡(θ)sin⁡(θ)cos⁡(θ))R_\theta = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}
    • Reflection across the xx-axis: (100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
    • Uniform Scaling by factor kk: (k00k)\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

    Step-by-Step Problem

    Problem: Let v⃗=(2−1)\vec{v} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} and matrix M=(0−231)M = \begin{pmatrix} 0 & -2 \\ 3 & 1 \end{pmatrix}. Find the transformed vector w⃗=Mv⃗\vec{w} = M\vec{v} and evaluate the dot product v⃗⋅w⃗\vec{v} \cdot \vec{w}.

    Solution

    1. Calculate w⃗=Mv⃗\vec{w} = M\vec{v}:
      w⃗=(0−231)(2−1)=((0)(2)+(−2)(−1)(3)(2)+(1)(−1))=(25)\vec{w} = \begin{pmatrix} 0 & -2 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} (0)(2) + (-2)(-1) \\ (3)(2) + (1)(-1) \end{pmatrix} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}
    2. Compute the dot product v⃗⋅w⃗\vec{v} \cdot \vec{w}:
      v⃗⋅w⃗=(2)(2)+(−1)(5)=4−5=−1\vec{v} \cdot \vec{w} = (2)(2) + (-1)(5) = 4 - 5 = -1

    Because v⃗⋅w⃗=−1≠0\vec{v} \cdot \vec{w} = -1 \neq 0, the transformed vector w⃗\vec{w} is not orthogonal to v⃗\vec{v}.