University:
Carleton College
Course:
MATH 111 | Introduction to Calculus
Academic year:

2019
Views:

190

Pages:

2
Author:

StellarSymphony

Section 1: Sketching Vectors Given the vector u = <2, 1>, the worksheet asks to sketch the following vectors: 2u = <4, 2> -v = <-3, 3> -2v = <6, -6> v - u = <1, -4> u + v = <5, -2> The worksheet provides a coordinate plane for sketching these vectors. Section 2: Expressing Vectors in Component Form Given the initial point P and terminal point Q, the task is to express the vector PQ in component form: P(3, 2), Q(10, 6) PQ = <10-3, 6-2> = <7, 4> P(-2, 4), Q(-8, -3) PQ = <-8-(-2), -3-4> = <-6, -7> Section 3: Vector Operations and Length Given vectors u = <-1, 4> and v = <3, -7>, the worksheet asks to find: -2u, u + v, 2v - u, |u|, |v|, and |u - v| Calculations are shown for each of these operations, including finding the magnitude (length) of vectors using the distance formula. Section 1: Vector Operations Given vectors u = <-1, 2> and v = <3, -1>, the worksheet asks to find: -2u, u + v, 2v - u, |u|, |v|, and |u - v| Calculations are shown for each of these operations, including finding the magnitude (length) of vectors using the distance formula. Section 2: Finding Components of Vectors Given the magnitude and direction of a vector, the task is to find its horizontal and vertical components. |v| = 5, θ = 2π/3 v₁ = |v| cosθ = 5 cos(2π/3) = -5/2 v₂ = |v| sinθ = 5 sin(2π/3) = 5√3/2 < -5/2, 5√3/2 > = -5/2 i + 5√3/2 j |v| = √3, θ = 240° v₁ = |v| cosθ = √3 cos(240°) = -√3/2 v₂ = |v| sinθ = √3 sin(240°) = -3√3/2 < -√3/2, -3√3/2 > = -√3/2 i - 3√3/2 j |v| = 6, θ = 225° v₁ = |v| cosθ = 6 cos(225°) = -3√2 v₂ = |v| sinθ = 6 sin(225°) = -3√2 < -3√2, -3√2 > = -3√2 i - 3√2 j |v| = 4, θ = π/2 v₁ = |v| cosθ = 4 cos(π/2) = 0 v₂ = |v| sinθ = 4 sin(π/2) = 4 < 0, 4 > = 0i + 4j Section 3: Finding Magnitude and Direction of Vectors Given a vector, the task is to find its magnitude and direction. <√2/2, -√2/2> Magnitude: √((√2/2)² + (-√2/2)²) = 1 Direction: tan⁻¹(-√2/2 / √2/2) = -π/4 v = 2i + 2j Magnitude: √(2² + 2²) = 2√2 Direction: tan⁻¹(2/2) = π/4 u = -i - √3j Magnitude: √((-1)² + (-√3)²) = 2 Direction: tan⁻¹(-√3/-1) = π/3 (-3√3, 3) Magnitude: √((-3√3)² + 3²) = 6 Direction: tan⁻¹(3/-3√3) = -π/6