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VectorsLinear algebraCollegeabout 0:43

Find the dot product and the angle between ⟨1, 2, 3⟩ and ⟨4, -5, 6⟩: animated lesson

A narrated, step-by-step vectors video lesson with captions. Final answer: u · v = 12, angle ≈ 68.562447°.

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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. The problem
Worked example
Find the dot product and the angle between ⟨1, 2, 3⟩ and ⟨4, -5, 6⟩.
2. Steps, part 1
1. $(1)(4) + (2)(-5) + (3)(6)$
  Multiply matching components
2. $u v = 4 - 10 + 18 = 12$
  Add the products
3. $|u| = \sqrt{14} = \sqrt{14} \approx 3.741657387,\quad |v| = \sqrt{77} = \sqrt{77} \approx 8.774964387$
  Find the magnitudes
4. $\cos\theta = \frac{u v}{|u||v|} = \frac{12}{32.83291} \approx 0.36548694 \Rightarrow \theta \approx 68.562447° \ (1.1966404\text{ rad})$
  Angle between the vectors
3. Steps, part 2
1. $\operatorname{comp}_v u = \frac{u v}{|v|} \approx 1.3675269,\quad \operatorname{proj}_v u = \frac{u v}{|v|^2}v = \langle \frac{48}{77}, -\frac{60}{77}, \frac{72}{77} \rangle$
  Scalar and vector projection of u onto v
4. The answer
$u v = 12$
u·v = 12
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, 2, 3⟩ and ⟨4, negative 5, 6⟩. 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 12, angle approximately 68.562447°. 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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