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VectorsLinear algebraCollegeabout 0:41

Find the dot product and the angle between ⟨1, 0, 1⟩ and ⟨0, 1, 1⟩: animated lesson

A narrated, step-by-step vectors video lesson with captions. Final answer: u · v = 1, angle ≈ 60°.

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 (0.3 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find the dot product and the angle between ⟨1, 0, 1⟩ and ⟨0, 1, 1⟩.
  2. 2. Steps, part 1
    1. (1)(0)+(0)(1)+(1)(1)(1)(0) + (0)(1) + (1)(1)
      Multiply matching components
    2. uv=0+0+1=1u v = 0 + 0 + 1 = 1
      Add the products
    3. ∣u∣=2=2≈1.414213562,∣v∣=2=2≈1.414213562|u| = \sqrt{2} = \sqrt{2} \approx 1.414213562,\quad |v| = \sqrt{2} = \sqrt{2} \approx 1.414213562
      Find the magnitudes
    4. cos⁡θ=uv∣u∣∣v∣=12≈0.5⇒θ≈60° (1.0471976 rad)\cos\theta = \frac{u v}{|u||v|} = \frac{1}{2} \approx 0.5 \Rightarrow \theta \approx 60° \ (1.0471976\text{ rad})
      Angle between the vectors
  3. 3. Steps, part 2
    1. comp⁡vu=uv∣v∣≈0.70710678,proj⁡vu=uv∣v∣2v=⟨0,12,12⟩\operatorname{comp}_v u = \frac{u v}{|v|} \approx 0.70710678,\quad \operatorname{proj}_v u = \frac{u v}{|v|^2}v = \langle 0, \frac{1}{2}, \frac{1}{2} \rangle
      Scalar and vector projection of u onto v
  4. 4. The answer
    uv=1u v = 1

    u·v = 1

  5. 5. Check
    • Computed exactly; the dot product is symmetric, so v·u gives the same value.

Transcript

The problem. Here is the problem: Find the dot product and the angle between ⟨1, 0, 1⟩ and ⟨0, 1, 1⟩. Dot product of two 3-dimensional vectors.

Steps, part 1. Multiply matching components. Add the products. Find the magnitudes. Angle between the vectors.

Steps, part 2. Scalar and vector projection of u onto v.

The answer. So the answer is u times v equals 1, angle approximately 60°. The dot product is positive, so the angle between the vectors is acute.

Check. Computed exactly; the dot product is symmetric, so v times u gives the same value.

About vectors

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

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