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VectorsLinear algebraCollegeabout 1:05

Find the magnitude and unit vector of ⟨2, -1, 3⟩: animated lesson

A narrated, step-by-step vectors video lesson with captions. Final answer: |u| = √14 ≈ 3.741657387; unit vector û = ⟨√14/7, -√14/14, 3√14/14⟩.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find the magnitude and unit vector of ⟨2, -1, 3⟩.
  2. 2. Step by step
    1. ∣u⃗∣2=22+(−1)2+32=14|\vec u|^2 = 2^2 + (-1)^2 + 3^2 = 14
      Square the components and add them
    2. ∣u⃗∣=14=14≈3.741657387|\vec u| = \sqrt{14} = \sqrt{14} \approx 3.741657387
      Take the square root for the magnitude
    3. u^=u⃗∣u⃗∣=(147, −1414, 31414)\hat u = \frac{\vec u}{|\vec u|} = \left(\frac{\sqrt{14}}{7},\ -\frac{\sqrt{14}}{14},\ \frac{3\sqrt{14}}{14}\right)
      Divide each component by the magnitude for the unit vector
    4. (0.5345224838)2+(−0.2672612419)2+(0.8017837257)2≈1(0.5345224838)^2 + (-0.2672612419)^2 + (0.8017837257)^2 \approx 1
      Check: the unit vector has length 1
  3. 3. The answer
    ∣u⃗∣=14≈3.741657387,u^=(147, −1414, 31414)|\vec u| = \sqrt{14} \approx 3.741657387,\quad \hat u = \left(\frac{\sqrt{14}}{7},\ -\frac{\sqrt{14}}{14},\ \frac{3\sqrt{14}}{14}\right)

    |u| = √14 ≈ 3.741657387; unit vector û = ⟨√14/7, -√14/14, 3√14/14⟩

  4. 4. Check
    • The unit vector's components square and add to 1.

Transcript

The problem. Here is the problem: Find the magnitude and unit vector of ⟨2, negative 1, 3⟩. Vector magnitude and unit vector using exact components.

Step by step. Square the components and add them. Take the square root for the magnitude. Divide each component by the magnitude for the unit vector. The denominators are rationalised, so each component is written with the square root on top. Check: the unit vector has length 1.

The answer. So the answer is |u| equals the square root of 14 approximately 3.741657387; unit vector û equals ⟨the square root of 14 over 7, negative the square root of 14 over 14, 3the square root of 14 over 14⟩. Vectors are added, subtracted and scaled one component at a time. The magnitude comes from the Pythagorean theorem, and dividing by it gives a unit vector of length 1 pointing the same way.

Check. The unit vector prime s components square and add to 1.

About vectors

Magnitude, unit vectors, dot products and angles between vectors.

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Find the magnitude and unit vector of ⟨2, -1, 3⟩ (Lesson)