Vectors `\vec {A}`, `\vec {b}` and `\vec {C}` are 4 units north, 3 units west and 8 units east, respectively. Describe carefully (a) `\vec {A}` x `\vec {B}` (b) `\vec {A}` x `\vec {C}` (c) `\vec {B}` x `\vec {C}`

[Ans: (a)12 units vertically up (b) 32 units vertically down (c) Zero]


Given:

`\vec {A}` = 4 units North
`\vec {B}` = 3 units West 
`\vec {C}` = 8 units East 

So, keeping in view the given directions of the vectors
Angle 𝜽 between `\vec {A}` and `\vec {B}` = 90⁰
Angle 𝜽 between `\vec {A}` and `\vec {C}` = 90⁰
Angle 𝜽 between `\vec {B}` and `\vec {C}` = 180⁰


To Find:

(a)  `\vec {A}` ｘ `\vec {B}`
(b)  `\vec {A}` ｘ `\vec {C}`
(c)  `\vec {B}` ｘ `\vec {C}`

Solution: 

(a)  `\vec {A}` ｘ `\vec {B}`

Here we have 

`\vec {A}` ｘ `\vec {B}` =  |`\vec {A}`|  |`\vec {B}`| Sin Ө `\hat {n}` 


 by putting values


`\vec {A}` ｘ `\vec {B}` =  4 ｘ 3 ｘ Sin 90⁰  `\hat {n}` 


[ Sin 90⁰ = 1 ]


`\vec {A}` ｘ `\vec {B}` =  4 ｘ 3 ｘ 1 ｘ 1 


`\vec {A}` ｘ `\vec {B}` =  12 units ------Ans


According to the right-hand rules the direction will be vertically upwards.


(b)  `\vec {A}` ｘ `\vec {C}`

Here we have 

`\vec {A}` ｘ `\vec {C}` =  |`\vec {A}`|  |`\vec {C}`| Sin Ө `\hat {n}` 


 by putting values


`\vec {A}` ｘ `\vec {C}` =  4 ｘ 8 ｘ Sin 90⁰  `\hat {n}` 

 
[ Sin 90⁰ = 1 ]

`\vec {A}` ｘ `\vec {C}` =  4 ｘ 8 ｘ 1ｘ 1 


`\vec {A}` ｘ `\vec {C}` =  32 units ------Ans


By right-hand rules, the direction will be vertically downwards.


(c)  `\vec {B}` ｘ `\vec {C}`

Here we have 

`\vec {B}` ｘ `\vec {C}` =  |`\vec {A}`|  |`\vec {C}`| Sin Ө `\hat {n}` 


 by putting values


`\vec {B}` ｘ `\vec {C}` =  3 ｘ 8 ｘ Sin 90⁰  `\hat {n}` 

 
[ Sin 180⁰ = 0 ]

`\vec {B}` ｘ `\vec {C}` =  3 ｘ 8 ｘ 0 ｘ 1 


`\vec {B}` ｘ `\vec {C}` =  0 units  -------Ans


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Numerical Problem 2.11
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List of 11th Physics Numerical Problems of Chapter - 2
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