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VectorsLinear algebraCollegeabout 0:43

Find the dot product and the angle between ⟨5, 2, -1⟩ and ⟨1, -3, 4⟩: animated lesson

A narrated, step-by-step vectors video lesson with captions. Final answer: u · v = -5, angle ≈ 100.31319°.

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 ⟨5, 2, -1⟩ and ⟨1, -3, 4⟩.
  2. 2. Steps, part 1
    1. (5)(1)+(2)(−3)+(−1)(4)(5)(1) + (2)(-3) + (-1)(4)
      Multiply matching components
    2. uv=5−6−4=−5u v = 5 - 6 - 4 = -5
      Add the products
    3. ∣u∣=30=30≈5.477225575,∣v∣=26=26≈5.099019514|u| = \sqrt{30} = \sqrt{30} \approx 5.477225575,\quad |v| = \sqrt{26} = \sqrt{26} \approx 5.099019514
      Find the magnitudes
    4. cos⁡θ=uv∣u∣∣v∣=−527.92848≈−0.17902872⇒θ≈100.31319° (1.7507955 rad)\cos\theta = \frac{u v}{|u||v|} = \frac{-5}{27.92848} \approx -0.17902872 \Rightarrow \theta \approx 100.31319° \ (1.7507955\text{ rad})
      Angle between the vectors
  3. 3. Steps, part 2
    1. comp⁡vu=uv∣v∣≈−0.98058068,proj⁡vu=uv∣v∣2v=⟨−526,1526,−1013⟩\operatorname{comp}_v u = \frac{u v}{|v|} \approx -0.98058068,\quad \operatorname{proj}_v u = \frac{u v}{|v|^2}v = \langle -\frac{5}{26}, \frac{15}{26}, -\frac{10}{13} \rangle
      Scalar and vector projection of u onto v
  4. 4. The answer
    uv=−5u v = -5

    u·v = -5

  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 ⟨5, 2, negative 1⟩ and ⟨1, negative 3, 4⟩. 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 negative 5, angle approximately 100.31319°. 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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