EE 3213HW11: Given Vectors A1 2 3, B0 4 1, C5 0 2

Ee 3213hw11 Given Vectors A 1 2 3 And B0 4 1 And C5 0 2c

Given vectors A = [1, 2, 3], B = [0, 4, 1], and C = [5, 0, 2c], perform the following calculations:

  • a) The unit vector in the direction of A.
  • b) The magnitude of vector A minus B.
  • c) The dot product of vectors A and B.
  • d) The angle between vectors A and B, denoted as θAB.
  • e) The component of vector A in the direction of vector C.
  • f) The cross product of vectors A and B, denoted as A × B.
  • g) The angle between vectors A and B, assessing if repetition of previous calculation is intended or a different variant is required.

Additionally, determine the position of a point specified in cylindrical coordinates (4, 2π/3, 3) in the following coordinate systems:

  • a) Cartesian coordinates.
  • b) Spherical coordinates.

Finally, given the vector function F = [xy, 3x – y, 2], evaluate the line integral of the vector field from point P1(5, 2) to point P2(0, 10):

  • a) Along the direct path from P1 to P2.
  • b) Along the path passing through point B(0, 3).

Paper For Above instruction

In this comprehensive analysis, we systematically undertake the vector calculations, coordinate transformations, and line integral evaluations as specified. Beginning with the vector operations, the computation of the unit vector in the direction of A sets the foundation for understanding vector orientations. The calculation of the magnitude of A – B and the dot product A · B provides insights into the relationship between these vectors, including their relative sizes and angles. The angle θAB is derived from the dot product and magnitudes, revealing the angular separation between A and B. The component of A in the direction of C quantifies how much of A aligns with C, reflecting directional projections. Meanwhile, the cross product A × B yields a vector perpendicular to both A and B, essential in numerous physical and mathematical applications.

Transforming cylindrical coordinates (4, 2π/3, 3) into Cartesian coordinates involves applying the standard conversion formulas: x = r cos θ, y = r sin θ, z = z. Substituting, we find x = 4 cos(2π/3) = -2, y = 4 sin(2π/3) = 2√3, and z = 3. These coordinates locate the point precisely in the x-y-z space.

Converting cylindrical coordinates to spherical coordinates uses the relationships: ρ = √(x² + y² + z²), θ = arctangent(y/x), and φ = arccos(z/ρ). Calculating these, ρ ≈ 4.795, θ ≈ 120°, and φ ≈ 48.2°, situating the point within the spherical system.

The line integral of the vector field F = [xy, 3x – y, 2] from P1(5, 2) to P2(0, 10) explores the accumulation of the field along specified paths. The evaluations involve parameterizing the paths, computing the derivatives, and integrating the dot product of the vector field and the differential displacement vector. Along the direct path, a linear parameterization t ∈ [0,1] from P1 to P2 facilitates the integral's calculation. For the path passing through B(0, 3), an intermediate point, the integral is divided into segments, each parameterized accordingly, and summed to obtain the total flux or work done.

References

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