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VectorsLinear algebraCollegeabout 0:42

Find the dot product and the angle between ⟨2, -1⟩ and ⟨1, 3⟩: animated lesson

A narrated, step-by-step vectors video lesson with captions. Final answer: u · v = -1, angle ≈ 98.130102°.

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 ⟨2, -1⟩ and ⟨1, 3⟩.
  2. 2. Steps, part 1
    1. (2)(1)+(−1)(3)(2)(1) + (-1)(3)
      Multiply matching components
    2. uv=2−3=−1u v = 2 - 3 = -1
      Add the products
    3. ∣u∣=5=5≈2.236067977,∣v∣=10=10≈3.16227766|u| = \sqrt{5} = \sqrt{5} \approx 2.236067977,\quad |v| = \sqrt{10} = \sqrt{10} \approx 3.16227766
      Find the magnitudes
    4. cos⁡θ=uv∣u∣∣v∣=−17.0710678≈−0.14142136⇒θ≈98.130102° (1.7126934 rad)\cos\theta = \frac{u v}{|u||v|} = \frac{-1}{7.0710678} \approx -0.14142136 \Rightarrow \theta \approx 98.130102° \ (1.7126934\text{ rad})
      Angle between the vectors
  3. 3. Steps, part 2
    1. comp⁡vu=uv∣v∣≈−0.31622777,proj⁡vu=uv∣v∣2v=⟨−110,−310⟩\operatorname{comp}_v u = \frac{u v}{|v|} \approx -0.31622777,\quad \operatorname{proj}_v u = \frac{u v}{|v|^2}v = \langle -\frac{1}{10}, -\frac{3}{10} \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 ⟨2, negative 1⟩ and ⟨1, 3⟩. Dot product of two 2-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 negative 1, angle approximately 98.130102°. The dot product is negative, so the angle between the vectors is obtuse.

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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