Find the angle between the two vectors, `\vec {A}` = 5`\hat {i}` + `\hat {j}` and `\vec {B}` = 2`\hat {i}` + 4`\hat {j}` (Ans: 52°)

Given:

`\vec {A}` = 5`\hat {i}` + `\hat {j}` 
`\vec {B}` = 2`\hat {i}` + 4`\hat {j}`


To Find:

Angle between the two vectors `\vec {A}` and `\vec {B}` = Ө = ?


Solution: 

We will solve this numerical by taking the dot product of the two vectors `\vec {A}` and `\vec {B}` given as 

`\vec {A}` . `\vec {B}` = |`\vec {A}`|  |`\vec {B}`| cos Ө

or

cos Ө =  `\frac {vec {A} . vec {B}}{|vec A| |vec B|}`

cos Ө =  `\frac {(5 hat {i} + hat {j}) . (2hat {i} + 4 hat {j})}{(sqrt {Ax^2 + Ay^2}) (sqrt {Bx^2 + By^2})}`

cos Ө =  `\frac {(5 ｘ 2) hat {i} . hat {i} + (5 ｘ 4) hat {i} . hat {j} + (1 ｘ 2) hat {j} . hat {i} + (1 ｘ 4) hat {j} . hat {j}}{(sqrt {5^2 + 1^2}) (sqrt {2^2 + 4^2})}`



cos Ө =  `\frac {10 + 4}{sqrt {25 + 1} ｘ sqrt {4 + 16}}`

cos Ө =  `\frac {14}{sqrt {26} ｘ sqrt {20}}`

cos Ө =  `\frac {14}{sqrt {26 ｘ 20}}`

cos Ө =  `\frac {14}{sqrt {520}}`

cos Ө =  `\frac {14}{22.804}`

cos Ө =  0.61

Ө = cos⁻¹ (0.61)

Ө = 52⁰

Thus, the angle Ө between the two vectors `\vec {A}` and `\vec {B}` is 52⁰

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Numerical Problem 2.7
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List of 11th Physics Numerical Problems of Chapter - 2
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