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<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
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    <title>Vectors in Two Dimensions</title>
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<body>

    <header>
        <h1>Vectors in Two Dimensions</h1>
    </header>

    <div class="content">
        <h2>1. Definition of a Vector</h2>
        <p>A vector is a mathematical entity that has both **magnitude** and **direction**.</p>

        <h2>2. Representation of Vectors</h2>
        <p>Vectors in two-dimensional space can be written in different ways:</p>

        <h3>Column Form:</h3>
        <p>A = [ x , y ]</p>

        <h3>Unit Vector Notation:</h3>
        <p>A = x<i>i</i> + y<i>j</i></p>

        <h2>3. Magnitude of a Vector</h2>
        <p>The length or magnitude of a vector **A = (x, y)** is calculated as:</p>
        <p>|A| = √(x² + y²)</p>

        <h2>4. Direction (Angle) of a Vector</h2>
        <p>The direction of a vector with respect to the x-axis is given by:</p>
        <p>θ = tan⁻¹(y / x)</p>

        <h2>5. Vector Addition</h2>
        <p>If A = (x₁, y₁) and B = (x₂, y₂), then:</p>
        <p>A + B = (x₁ + x₂, y₁ + y₂)</p>

        <h2>6. Vector Subtraction</h2>
        <p>A - B = (x₁ - x₂, y₁ - y₂)</p>

        <h2>7. Scalar Multiplication</h2>
        <p>If k is a scalar and A = (x, y), then:</p>
        <p>kA = (kx, ky)</p>

        <h2>8. Dot Product of Two Vectors</h2>
        <p>The dot product of two vectors A and B is calculated as:</p>
        <p>A • B = x₁x₂ + y₁y₂</p>
        <p>or equivalently,</p>
        <p>A • B = |A| |B| cosθ</p>

        <h2>9. Cross Product in 2D</h2>
        <p>In two dimensions, the cross product is given by:</p>
        <p>A × B = (x₁y₂ - y₁x₂)</p>

        <h2>10. Unit Vector</h2>
        <p>A unit vector in the direction of A is given by:</p>
        <p>Unit vector Â = A / |A|</p>

        <h2>11. Projection of a Vector</h2>
        <p>The projection of vector A onto vector B is given by:</p>
        <p>Proj<sub>B</sub> A = (A • B) / |B|² * B</p>

        <h2>12. Vector Rotation</h2>
        <p>A vector can be rotated by an angle θ using:</p>
        <p>A' = (x cosθ - y sinθ, x sinθ + y cosθ)</p>

        <h2>13. Vector Components</h2>
        <p>A vector A at an angle θ can be broken into components:</p>
        <p>Ax = |A| cosθ</p>
        <p>Ay = |A| sinθ</p>

        <h2>14. Position Vector</h2>
        <p>The position vector of a point (x, y) from the origin is:</p>
        <p>R = xi + yj</p>

        <h2>15. Velocity and Acceleration Vectors</h2>
        <p>For a moving object with position vector R(t):</p>
        <p>Velocity: V = dR/dt</p>
        <p>Acceleration: A = d²R/dt²</p>

        <h2>16. Work Done by a Force</h2>
        <p>Work W done by a force F moving an object along displacement d:</p>
        <p>W = F • d = |F| |d| cosθ</p>

        <h2>17. Torque</h2>
        <p>Torque τ produced by a force F at a position vector r:</p>
        <p>τ = r × F</p>

        <h2>18. Vector Equation of a Line</h2>
        <p>The equation of a line passing through A with direction B:</p>
        <p>r = A + tB</p>

    </div>

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        <p>&copy; 2025 Yoseph Feyisa Wegi | All Rights Reserved.</p>
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