I Need Someone Who Can Do These Questions Correctly 1 A Disp

I Need Somone Who Can Do This Quesstions Correctly 1 A Displacement V

I need someone who can do this questions correctly:

1- A displacement vector in the xy plane is 52 m long and directed at an angle θ = 22° in the figure. Determine (a) the x component and (b) the y component of the vector. (a) Number (b) Number

2- In the figure, a heavy piece of machinery is raised by sliding it a distance d = 17.7 m along a plank oriented at an angle θ = 37.8° to the horizontal. How far is it moved (a) vertically and (b) horizontally? (a) Number (b) Number

3- Find the (a) x, (b) y, and (c) z components of the sum of the displacements, given their components along the axes: cₓ = 3.9, cᵧ = -4.3, c_z = -6.5; dₓ = 4.3, dᵧ = -1.9, d_z = 3.3. (a) Number (b) Number (c) Number

4- For the vectors and , give in (a) unit-vector notation, and as (b) a magnitude and (c) an angle (relative to in the range of (-180°, 180°]). Now give in (d) unit-vector notation, and as (e) a magnitude and (f) an angle (relative to in the range of (-180°, 180°]). (a) Number (b) Number (c) Number (d) Number (e) Number (f) Number

Paper For Above instruction

The set of questions provided spans fundamental concepts in vector analysis and application within physics, particularly in the context of displacement, resolution into components, and vector addition. Each of these problems showcases core skills necessary for understanding motion and the representation of physical quantities in a multidimensional space.

Question 1: Components of a Displacement Vector

The first problem refers to a displacement vector in the xy-plane, given its magnitude and direction angle. To find the components, the standard approach involves trigonometric functions: the x-component is determined via the cosine of the angle (since it is adjacent to the angle in a right triangle), while the y-component is determined via the sine of the angle (opposite to the angle). Mathematically, these are expressed as:

• x component (A): x = |V| cos(θ)
• y component (B): y = |V| sin(θ)

Substituting the given values (|V| = 52 m, θ = 22°), we get:

• x = 52 × cos(22°) ≈ 52 × 0.9272 ≈ 48.2 m
• y = 52 × sin(22°) ≈ 52 × 0.3746 ≈ 19.5 m

This straightforward application of trigonometry helps resolve vectors into their orthogonal components, essential for analyzing motion or forces in each direction.

Question 2: Horizontal and Vertical Displacements Along a Sloped Plank

Here, the problem involves a piece of machinery moved along a plank inclined at an angle θ = 37.8°, with the length of the movement d = 17.7 meters. The goal is to find the horizontal and vertical displacements.

The horizontal and vertical components are calculated similarly to the previous problem, considering the nature of the movement along the inclined surface:

• Horizontal displacement (a): h = d × cos(θ)
• Vertical displacement (b): v = d × sin(θ)

Calculating with the given values:

• h = 17.7 × cos(37.8°) ≈ 17.7 × 0.793 ≈ 14.05 m
• v = 17.7 × sin(37.8°) ≈ 17.7 × 0.607 ≈ 10.75 m

These components represent how far the object moves horizontally and vertically, which would be crucial for logistical planning or mechanical analysis.

Question 3: Adding Three-Dimensional Displacements

In the third question, the vector addition involves three components along x, y, and z axes. The sum of two vectors in component form is simply the sum of their respective components:

• x component: cₓ + dₓ = 3.9 + 4.3 = 8.2
• y component: cᵧ + dᵧ = -4.3 + -1.9 = -6.2
• z component: c_z + d_z = -6.5 + 3.3 = -3.2

The resultant vector components are therefore (8.2, -6.2, -3.2) meters. To further analyze these, the magnitude is calculated using the Euclidean distance formula:

|R| = √(x² + y² + z²) ≈ √(8.2² + (-6.2)² + (-3.2)²) ≈ √(67.24 + 38.44 + 10.24) ≈ √115.92 ≈ 10.78 m

Directionally, the orientation can be described by computing the angles relative to axes, often referencing the x-axis or a specific plane to provide meaningful directional information.

Question 4: Vector Representation and Magnitude-Angle Relationships

This multi-part problem emphasizes the conversion between different vector representations:

• (a) Unit-vector notation: writing vectors as combinations of î, ĵ, k̂ with scalar coefficients, for example, V = Vₓî + Vᵧĵ + V_zk̂
• (b) Magnitude: calculated as √(Vₓ² + Vᵧ² + V_z²)
• (c) Angle: determined through inverse tangent functions (e.g., θ = tan⁻¹(Vᵧ / Vₓ)) considering the signs of each component to find the correct quadrant, within (-180°, 180°).

The problem requires expressing vectors in different forms, calculating magnitudes, and establishing their angles relative to reference axes. These skills are fundamental in vector kinematics and dynamics, enabling a clear understanding of motion directions and magnitudes in three-dimensional space.

By mastering these geometric and algebraic methods, one can analyze complex vector relationships, which underpin a wide range of applications in physics and engineering, from robotics to aerospace navigation.

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