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VectorsLinear algebraCollegeabout 0:39

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

A narrated, step-by-step vectors video lesson with captions. Final answer: u · v = 0, angle ≈ 90°.

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 ⟨3, 4⟩ and ⟨-4, 3⟩.
  2. 2. Steps, part 1
    1. (3)(−4)+(4)(3)(3)(-4) + (4)(3)
      Multiply matching components
    2. uv=−12+12=0u v = -12 + 12 = 0
      Add the products
    3. ∣u∣=25=5,∣v∣=25=5|u| = \sqrt{25} = 5,\quad |v| = \sqrt{25} = 5
      Find the magnitudes
    4. cos⁡θ=uv∣u∣∣v∣=025≈0⇒θ≈90° (1.5707963 rad)\cos\theta = \frac{u v}{|u||v|} = \frac{0}{25} \approx 0 \Rightarrow \theta \approx 90° \ (1.5707963\text{ rad})
      Angle between the vectors
  3. 3. Steps, part 2
    1. comp⁡vu=uv∣v∣≈0,proj⁡vu=uv∣v∣2v=⟨0,0⟩\operatorname{comp}_v u = \frac{u v}{|v|} \approx 0,\quad \operatorname{proj}_v u = \frac{u v}{|v|^2}v = \langle 0, 0 \rangle
      Scalar and vector projection of u onto v
  4. 4. The answer
    uv=0u v = 0

    u·v = 0

  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 ⟨3, 4⟩ and ⟨-4, 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 0, angle approximately 90°. The dot product is 0, so the vectors are orthogonal (perpendicular).

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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