Vector Components

Trigonometry · Axiom Academy

Breaking vectors into horizontal and vertical components using trigonometry— the foundation for analyzing forces, velocities, and motion in two dimensions. A vector is a quantity with both magnitude (size) and direction . Unlike scalars (which only have magnitude), vectors tell us not just "how much" but also "which way." Magnitude: The length of the vector (denoted |v| or ||v||) Direction: The angle θ measured from the positive x-axis Representation: An arrow pointing from origin to endpoint Any vector can be decomposed into two perpendicular components: one along the x-axis (horizontal) and one along the y-axis (vertical). This is like asking: "How much does this vector move me to the right, and how much up?" To find the component values, we use the trigonometric functions cosine and sine. These relate the angle to the ratios of the triangle's sides. vₓ = |v| cos(θ) — Horizontal component vᵧ = |v| sin(θ) — Vertical component Let's find the components of a vector with magnitude 50 at an angle of 30° above the horizontal. Horizontal: vₓ = 50 × cos(30°) = 50 × 0.866 ≈ 43.3 Vertical: vᵧ = 50 × sin(30°) = 50 × 0.5 = 25 Result: The vector moves us 43.3 units right and 25 units up

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