Vector Direction and Unit Vectors

Direction of a Vector

  • Direction refers to the angle from a horizontal line.
  • Vectors can be moved to start at the origin to think in terms of quadrants.
  • To find the direction of a vector, determine the angle it forms with the horizontal.

Finding the Angle

  • Given xx component = 8 and yy component = 6.
  • Using trigonometric functions:
    • tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
    • tan⁡(θ)=68\tan(\theta) = \frac{6}{8}
  • To find θ\theta (the angle), use the inverse tangent:
    • θ=tan⁡−1(68)\theta = \tan^{-1}(\frac{6}{8})

Calculator Settings

  • Ensure the calculator is in degree mode.
  • Calculate the inverse tangent of 68\frac{6}{8}.
  • θ≈36.87\theta \approx 36.87 degrees.

General Formula and Quadrant Considerations

  • General formula: tan⁡−1(yx)\tan^{-1}(\frac{y}{x})
  • Inverse tangent gives angle measures in quadrants one and four.
  • Modify θ\theta based on the quadrant.

Example 1

  • Vector with components: right 3, up 4.
  • Angle measure is from the horizontal to the vector.
  • θ=tan⁡−1(43)\theta = \tan^{-1}(\frac{4}{3})
  • θ≈53.13\theta \approx 53.13 degrees.

Example 2

  • Vector: −8i+4j-8i + 4j which is equivalent to (−8,4)(-8, 4).
  • The vector goes left 8 and up 4.
  • The angle is the angle from the horizontal line.
  • Calculate tan⁡−1(4−8)\tan^{-1}(\frac{4}{-8}).
  • The calculator gives −26.57-26.57 degrees, which is in quadrant four.
  • To find the correct angle in quadrant two, add 180 degrees:
    • −26.57+180=153.43-26.57 + 180 = 153.43 degrees.

Finding Component Form

  • Given the vector's magnitude (hypotenuse) and angle.
  • To find the xx component (adjacent):
    • cos⁡(θ)=xhypotenuse\cos(\theta) = \frac{x}{\text{hypotenuse}}
    • x=hypotenuse⋅cos⁡(θ)x = \text{hypotenuse} \cdot \cos(\theta)
  • To find the yy component (opposite):
    • sin⁡(θ)=yhypotenuse\sin(\theta) = \frac{y}{\text{hypotenuse}}
    • y=hypotenuse⋅sin⁡(θ)y = \text{hypotenuse} \cdot \sin(\theta)

Example Calculation

  • Given: hypotenuse = 14, angle = 55 degrees.
  • x=14⋅cos⁡(55)≈8.03x = 14 \cdot \cos(55) \approx 8.03
  • y=14⋅sin⁡(55)≈11.47y = 14 \cdot \sin(55) \approx 11.47

General Formulas for Component Form

  • x=magnitude⋅cos⁡(θ)x = \text{magnitude} \cdot \cos(\theta)
  • y=magnitude⋅sin⁡(θ)y = \text{magnitude} \cdot \sin(\theta)

Unit Vectors

  • A unit vector is a vector with a magnitude of one.

Finding a Unit Vector

  • Take a vector and scale it to have a magnitude of one.

Example: Vector with Magnitude 5

  • Given vector with magnitude = 5.
  • If the magnitude isn't given, use the Pythagorean theorem to find the length.