What is the relationship between vectors A, B, and C, given that vector A points in the negative x direction, vector B points at an angle of 30 degrees above the positive x-axis, and vector C has a magnitude of 17 m and points in a direction of 420 degrees?

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Understanding the Relationship Between Vectors A, B, and C

To analyze the relationship between the three given vectors, let’s break them down:

1. Vector A

Vector A points in the negative x-direction. This can be represented in standard Cartesian coordinates as:

 A = (-a, 0)

where ‘a’ is the magnitude of vector A.

2. Vector B

Vector B has an angle of 30 degrees above the positive x-axis. This means it has components in both the x and y directions. It can be expressed as:

 B = (b * cos(30°), b * sin(30°))

where ‘b’ is the magnitude of vector B. By substituting the values:

  • cos(30°) = √3/2
  • sin(30°) = 1/2

Thus, it becomes:

 B = (b * √3/2, b * 1/2)

3. Vector C

Vector C has a magnitude of 17 m and points in a direction of 420 degrees. Since angles in standard positioning are measured counterclockwise from the positive x-axis, an angle of 420° is equivalent to:

 420° - 360° = 60°

Vector C can also be broken down into its x and y components as follows:

 C = (17 * cos(60°), 17 * sin(60°))

Substituting the trigonometric values:

  • cos(60°) = 1/2
  • sin(60°) = √3/2

This results in:

 C = (17 * 1/2, 17 * √3/2) = (8.5, 14.72)

Conclusion

Vectors A, B, and C can now be compared based on their components. Vector A points leftward, vector B rises at a 30 degree angle above the x-axis, while vector C points upwards at 60 degrees (or 420 degrees equivalently) with a moderate magnitude.

This analysis provides a clearer understanding of how these vectors behave relative to one another in a two-dimensional plane.

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