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How to Rewrite a Vector in Component Form?

Published in Vectors 3 mins read

Rewriting a vector in component form involves expressing the vector in terms of its projections along orthogonal axes (typically x and y in 2D, and x, y, and z in 3D).

Here's a breakdown of how to do it:

1. Understanding Component Form:

A vector in component form is represented as an ordered set of numbers, each representing the vector's magnitude along a specific axis. For example:

  • 2D: v = (vx, vy) where vx is the x-component and vy is the y-component.
  • 3D: v = (vx, vy, vz) where vx is the x-component, vy is the y-component, and vz is the z-component.

2. Finding the Components:

The method for finding the components depends on the information you have about the vector:

  • Magnitude and Direction (Angle): If you know the vector's magnitude (||v|| or |v|) and the angle (Θ) it makes with a reference axis (usually the positive x-axis), you can use trigonometry:

    • vx = ||v|| * cos(Θ)
    • vy = ||v|| * sin(Θ)

    In 3D, you'll need two angles (e.g., azimuth and elevation) to determine the x, y, and z components. The calculations become more complex.

  • Given Two Points: If you know the initial point (P1 = (x1, y1)) and terminal point (P2 = (x2, y2)) of the vector, then:

    • vx = x2 - x1
    • vy = y2 - y1

    In 3D, with P1 = (x1, y1, z1) and P2 = (x2, y2, z2):

    • vx = x2 - x1
    • vy = y2 - y1
    • vz = z2 - z1

3. Writing the Vector in Component Form:

Once you have calculated the components, simply write the vector as an ordered set:

  • 2D: v = (vx, vy)
  • 3D: v = (vx, vy, vz)

Example:

Let's say a vector v has a magnitude of 5 and makes an angle of 30 degrees with the x-axis.

  1. Find the x-component:
    vx = 5 * cos(30°) = 5 * (√3 / 2) ≈ 4.33

  2. Find the y-component:
    vy = 5 * sin(30°) = 5 * (1 / 2) = 2.5

  3. Write in component form:
    v = (4.33, 2.5)

Summary:

To rewrite a vector in component form, determine the vector's projections onto the coordinate axes. If you know the magnitude and direction, use trigonometric functions (sine and cosine). If you know the initial and terminal points, subtract the coordinates of the initial point from the coordinates of the terminal point. Finally, express the vector as an ordered set of these components.

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