University:
University of North Georgia
Course:
MATH 1113 | Precalculus
Academic year:

2024
Views:

467

Pages:

2
Author:

StellarStrategist

Vectors Draw the vector from P₁ to P₂. Then find the vector in component form with the initial point P₁ and terminal point P₂. Finally, draw the vector in component form. P₁(-2, 1) and P₂(5, -6) Vector: <7, -7> P₁(6, -1) and P₂(4, 3) Vector: <-2, 4> Find the magnitude and direction for the given vector. Draw the vector to help decide the direction! Estimate the magnitude to the nearest tenth. Estimate the direction to the nearest tenth of a degree. v = <5, -12> Magnitude: √(5² + (-12)²) = 13 Direction: tan⁻¹(-12/5) ≈ -67.4° Q4: 360° - 67.4° = 292.6° v = <-4, -7> Magnitude: √((-4)² + (-7)²) = √65 ≈ 8.1 Direction: tan⁻¹(-7/-4) ≈ 60.3° Q3: 180° + 60.3° = 240.3° vectors and calculations Given u = 4i - j and v = -6i + 5j, find the following: 15. 2u - 3v = 2(4i - j) - 3(-6i + 5j) = 8i - 2j + 18i - 15j = 26i - 17j 16. -5v = -5(-6i + 5j) = 30i - 25j 17. ||v|| = √((-6)² + 5²) = √61 18. u + 2v = (4i - j) + 2(-6i + 5j) = 4i - j - 12i + 10j = -8i + 9j 19. ||u + 2v|| = √((-8)² + 9²) = √145 ≈ 12.0 20. (2u - v) ∙ (-3i + 2.5j) = (8i - 2j - (-6i + 5j)) ∙ (-3i + 2.5j) = (14i - 7j) ∙ (-3i + 2.5j) = -42 + 17.5 = -24.5 Draw vectors u and v on the same coordinate plane. Then, find the dot product of u and v, ||u|| and ||v||. Finally, find the measure of the angle between the two vectors. 21. u = (4, -9); v = (2, 6) ||u|| = √(4² + (-9)²) = √97 ||v|| = √(2² + 6²) = √40 u ∙ v = 4(2) + (-9)(6) = -46 θ = cos⁻¹(-46 / (√97 * √40)) ≈ 158.2° (Obtuse angle) 22. u = (-3, 4); v = (4, 0) ||u|| = √((-3)² + 4²) = 5 ||v|| = √(4²) = 4 u ∙ v = (-3)(4) + 4(0) = -12 θ = cos⁻¹(-12 / (5 * 4)) = 90° (Right angle) 23. u = (1, 2); v = (-3, 7) ||u|| = √(1² + 2²) = √5 ||v|| = √((-3)² + 7²) = √58 u ∙ v = 1(-3) + 2(7) = 11 θ = cos⁻¹(11 / (√5 * √58)) ≈ 48.9° (Acute angle)