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Stolb23 [73]
2 years ago
6

The angle θ between 5i-j+k & 2i-j+k is​

Mathematics
1 answer:
Maksim231197 [3]2 years ago
7 0

Step-by-step explanation:

Let,

\sf \vec{a} = 5 \hat{i} -  \hat{j}  +  \hat{k} \\  \therefore \: \sf \:   | \vec{a}|  =  \sqrt{ {5}^{2}  +  {( - 1)}^{2}  +  {1}^{2} }  \\  =  \sqrt{25 + 1 + 1}  \\  =  \sqrt{27}  \\  \\  \sf \vec{b} = 2\hat{i} -  \hat{j}  +  \hat{k} \\ \therefore \: \sf \:  | \vec{b}|  =  \sqrt{ {2}^{2}  +  {( - 1)}^{2}  +  {1}^{2} }  \\  =  \sqrt{4 + 1 + 1}  \\  =  \sqrt{6}  \\  \\\sf \: \vec{a}. \vec{b} = (5 \hat{i} -  \hat{j}  +  \hat{k}).(2\hat{i} -  \hat{j}  +  \hat{k}) \\  = 5 \times 2 + ( - 1) \times ( - 1) + 1 \times 1 \\  = 10 + 1 + 1 \\  = 12 \\  \\  \sf \: angle \: between \:  \vec{a} \: and \:  \vec{b} \:  = \theta \\  \\  \: so \\  \sf \vec{a}. \vec{b} =  | \vec{a}| . | \vec{b}| cos\theta \\  =  > \sf \:  cos \theta \:  =  \frac{ \vec{a}. \vec{b}}{ | \vec{a}| . | \vec{b}| }  \\  =  > cos \theta =  \frac{12}{ \sqrt{27}  \times  \sqrt{6}  }  = 0.94 \\  =  >  \theta =  {cos}^{ - 1} (0.94) \\  =  >   \green{\theta = 19.47 ^{ \circ} }

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