PlanetPhysics/Dot Product Example

Examples involving the [[../DotProduct/|dot product]]:

(1) Calculate $𝐀\cdot 𝐁$ with

$𝐀=5\widehat{𝐢}+2\widehat{𝐣}-\widehat{𝐤}$ \\ $𝐁=3\widehat{𝐢}-3\widehat{𝐣}+3\widehat{𝐤}$ \\

answer:

$𝐀\cdot 𝐁=(5)(3)+(2)(-3)+(-1)(+3)$ \\ $𝐀\cdot 𝐁=15-6-3=6$ \\

(2) Find the angle between the above [[../Vectors/|vectors]].

answer:

We know their dot product, so we just need to calculate their [[../AbsoluteMagnitude/|magnitudes]] $\left|𝐀\right|=\sqrt{A_x^2+A_y^2+A_z^2}=\sqrt{5^2+2^2+(-1{)}^2}$ \\ $\left|𝐀\right|=\sqrt{25+4+1}=\sqrt{30}=5.48$ \\ \\

$\left|𝐁\right|=\sqrt{3^2+(-3{)}^2+(3{)}^2}$ \\ $\left|𝐁\right|=\sqrt{9+9+9}=\sqrt{27}=5.2$ \\ \\

Finally

$𝐀\cdot 𝐁=\left|𝐀\right|\left|𝐁\right|\cos \theta$

$\theta =co{s}^{-1}\left(\frac{𝐀\cdot 𝐁}{\left|𝐀\right|\left|𝐁\right|}\right)$ \\

$\theta =co{s}^{-1}\left(\frac{6}{(5.48)(5.2)}\right)$ \\ $\theta =co{s}^{-1}(0.21)=77.9^o$