Ma1310 Week 7 Solving Vectors Lab
Ma1310 Week 7 Solving Vectorsthis Lab Requires You To Use Magnitude
Ma1310: Week 7 Solving Vectors This lab requires you to: · Use magnitude and direction to show vectors are equal. · Visualize scalar multiplication, vector addition, and vector subtraction as geometric vectors. · Represent vectors in the rectangular coordinate system. · Perform operations with vectors in terms of i and j . · Find the unit vector in the direction of v . · Write a vector in terms of its magnitude and direction. · Solve applied problems involving vectors. · Find the dot product of two vectors. · Find the angle between two vectors. · Use the dot product to determine if two vectors are orthogonal. Answer the following questions to complete this lab: 1. What is a directed line segment? 2. What are equal vectors? 3. Let v be a vector from initial point P 1 = (–11, –12) to terminal point P 2 = (2, 5). Write v in terms of i and j . 4. Let u = 8 i – 6 j and v = –5 i + 8 j . Find the vector u + v . 5. Find the unit vector that has the same direction as the vector v = –12 j . 6. A Boeing 737 aircraft maintains a constant speed of 500 miles per hour headed due south. The jet stream is 80 miles per hour in the northeasterly direction. a. Express the velocity va of the 737 relative to the air, and the velocity vw of the jet stream in terms of i and j . b. Find the velocity vg of the 737 relative to the ground. (Hint: vg = va + vw ) c. Find the actual speed of the 737 relative to the ground. (Hint: ) 7. The components of v = 260 i + 300 j* represent the respective number of gallons of regular and premium gas sold at a gas station. The components of w = 2.50 i + 2.65 j represent the respective prices per gallon for each kind of gas. Find the dot product and describe what the answer means in practical terms. 8. A force of 80 pounds on a rope is used to pull a box up a ramp inclined at 100 from the horizontal. The rope forms an angle of 330 with the horizontal. How much work is done pulling the box 25 feet along the ramp?
Paper For Above instruction
Introduction
Vectors are fundamental concepts in physics and mathematics, representing quantities that have both magnitude and direction. Understanding vectors, their properties, and their applications is essential for solving real-world problems involving motion, forces, and various physical phenomena. This paper aims to explore key aspects of vectors, including their representation, operations, and applications, as outlined in the Week 7 laboratory exercise.
Directed Line Segment and Equal Vectors
A directed line segment is a geometric figure that has a starting point and an endpoint, with the segment indicating the direction from the initial point to the terminal point. It inherently includes both magnitude—length of the segment—and direction. Equal vectors are vectors that have the same magnitude and direction, regardless of their initial points. These vectors are considered to be identical in the vector space because their effect, when applied, is the same.
Vector Representation and Operations
Given the points P₁ = (–11, –12) and P₂ = (2, 5), the vector v from P₁ to P₂ can be computed by subtracting the coordinates component-wise:
v = P₂ – P₁ = (2 – (–11), 5 – (–12)) = (13, 17).
Expressed in terms of i and j, this becomes v = 13i + 17j.
For vectors u = 8i – 6j and v = –5i + 8j, their sum u + v is computed as:
u + v = (8 + (–5))i + (–6 + 8)j = 3i + 2j.
Scalar multiplication, vector addition, and subtraction are visually represented as geometric translation of vectors in the coordinate system, illustrating how combined vectors behave under these operations.
Unit Vectors and Magnitude
A unit vector is a vector of magnitude 1 that points in the same direction as the original vector. For the vector v = –12j, its magnitude is |v| = 12, and its unit vector is obtained by dividing each component by the magnitude:
unit_v = (0i – (12/12)j) = 0i – 1j = –j.
Application to Aviation: Velocity Vectors
In the context of aviation, vectors can describe velocity and displacement. Suppose a Boeing 737 maintains a speed of 500 mph heading south, which corresponds to the vector va = 0i – 500j. The jet stream's speed of 80 mph toward the northeast can be expressed in components by resolving it into i and j components, considering the angle of the stream. For instance, if the stream is directed at 45°, the components are:
vw = 80cos(45°)i + 80sin(45°)j ≈ 56.57i + 56.57j.
The resultant ground velocity vg is obtained by vector addition:
vg = va + vw = (0 + 56.57)i + (–500 + 56.57)j ≈ 56.57i – 443.43j.
The magnitude, representing the actual speed over the ground, is:
|vg| = √(56.57² + (–443.43)²) ≈ 447.55 mph.
This calculation illustrates how wind vectors influence aircraft speed and direction—a critical aspect of navigation and aviation safety.
Dot Product and Practical Applications
The dot product of two vectors v and w, given by v · w = v₁w₁ + v₂w₂, measures the extent to which vectors point in the same direction. For example, if v = 260i + 300j and w = 2.50i + 2.65j, then:
v · w = (260)(2.50) + (300)(2.65) = 650 + 795 = 1445.
This positive value indicates that the vectors have a component in common, which, in the context of the gas station, could represent the total value of sales for a combination of gases.
The angle θ between vectors v and w can be determined using the formula:
cosθ = (v · w) / (|v| |w|),
where |v| and |w| are the magnitudes. Calculations show that the vectors are not orthogonal, but rather have a certain degree of alignment.
Work and Force
Work done by a force is calculated as W = Fd cosθ, where F is the magnitude of the force, d is the displacement along the direction of the force, and θ is the angle between the force and displacement vectors. For example, pulling a box up a ramp with a force of 80 pounds along an inclined plane of 25 feet at an angle of 33° with the horizontal involves calculating the component of force along the ramp:
W = 80 × 25 × cos(33°) ≈ 80 × 25 × 0.8387 ≈ 1677.4 foot-pounds.
This quantifies the work involved in moving objects against gravitational and frictional forces.
Conclusion
The study of vectors encompasses representation, operations, and real-world applications such as navigation, force analysis, and economic calculations. Including magnitude, direction, dot product, and scalar multiplication deepens understanding and provides tools for solving complex problems. Accurate vector analysis is crucial in diverse fields like physics, engineering, economics, and transportation, illustrating the importance of mastery in vector concepts.
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