Numford
Vectors

Unit vector

The unit vector of 3i + 4j is 0.6i + 0.8j, and its magnitude is 1.

Vectors

Drag the glowing tips, or type values. Angles are measured from the x axis, and θ is the angle from A to B. Pick a result card to see it built in the picture.

Angle between14.25°cos θ = A·B / AB
Unit vectors0.6i + 0.8jB = 0.38i + 0.92j

Vector A of size 5 and vector B of size 13 with 14.25 degrees between them. The resultant is 17.89, 10.3 degrees from A. The dot product is 63 and the cross product has size 16.

A B
Show the formulas
Resultant
R = √(A² + B² + 2AB cos θ) = √(5² + 13² + 2 × 5 × 13 × cos 14.25°) = 17.89
Direction
tan β = B sin θ / (A + B cos θ) = 13 × sin 14.25° / (5 + 13 × cos 14.25°), so β = 10.3° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(5² + 13² - 2 × 5 × 13 × cos 14.25°) = 8.25
Dot product
A·B = AB cos θ = 5 × 13 × cos 14.25° = 63
Cross product
|A×B| = AB sin θ = 5 × 13 × sin 14.25° = 16
Components
A·B = AxBx + AyBy + AzBz = 3 × 5 + 4 × 12 + 0 × 0 = 63
Determinant
A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 16k
Unit vector
 = A / |A| = 0.6i + 0.8j
Working

Step by step

  1. Given A = 3i + 4j and B = 5i + 12j.
  2. |A| = √(9 + 16) = 5
  3. Â = A / |A| = (3i + 4j) / 5 = 0.6i + 0.8j
  4. |B| = √(25 + 144) = 13, so B̂ = 0.38i + 0.92j.
  5. Check: √(0.36 + 0.64) = 1. A unit vector always has magnitude 1.
Derivation

Why it works

  1. A unit vector keeps the direction of a vector and has length 1.
  2. To get it, divide the vector by its own magnitude: Â = A / |A|.
  3. In components, divide each one: Â = (Ax i + Ay j + Az k) / √(Ax² + Ay² + Az²).
  4. Check it: the magnitude of  is √(Ax² + Ay² + Az²) / |A| = 1.
  5. Any vector is its magnitude times its unit vector: A = |A| Â.
Questions

Unit vector: quick answers

How do you find a unit vector?

Divide the vector by its magnitude. In components, divide each one by √(Ax² + Ay² + Az²).

What is the unit vector of 3i + 4j?

|A| = 5, so  = 0.6i + 0.8j.

Are i, j and k unit vectors?

Yes. They are unit vectors along the x, y and z axes, and each is perpendicular to the other two.

Try your own vectors