dot product examples

(Example)

Examples involving the dot product:

(1) Calculate A ⋅ B with

A = 5î + 2ĵ −k
B = 3î − 3ĵ + 3k

answer:

A ⋅ B = (5)(3) + (2)(−3) + (−1)(+3)
A ⋅ B = 15 − 6 − 3 = 6

(2) Find the angle between the above vectors.

answer:

We know their dot product, so we just need to calculate their magnitudes |A | = ∘ --------------
A2x + A2y + A2z = ∘ ----------------
52 + 22 + (− 1)2
|A | = ----------
√ 25 + 4 + 1 = √ ---
30 = 5.48

|B | = ∘ -----------------
32 + (− 3 )2 + (3)2
|B | = √ ---------
9 + 9 + 9 = √---
27 = 5.2

Finally

A ⋅ B = |A | |B| cos 𝜃

𝜃 = cos⁻¹ ( )
A-⋅B--
|A||B |

𝜃 = cos⁻¹ ( )
---6----
(5.48)(5.2)
𝜃 = cos⁻¹(0.21) = 77.9^o

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