If a , b , c are three non-zero vectors and n ^ is a unit vector perpendicular to c such that a = b - n ^ , ( 0 ) and b · c = 12 , then | c × ( a × b ) | is equal to [2023]

Question

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = α \vec{b} - \hat{n}, (α \neq 0)$ and $\vec{b} \cdot \vec{c} = 12,$ then $| \vec{c} \times (\vec{a} \times \vec{b}) |$ is equal to [2023]

Smart Achievers · Accepted Answer

(4)

We have, $| \hat{n} |$ $= 1$

Given, $\hat{n} ⊥ \vec{c},$ $\vec{a} = α \vec{b} - \hat{n}$ and $\vec{b} \cdot \vec{c} = 12$

Now, $\vec{a} \times \vec{b} = (α \vec{b} - \hat{n}) \times \vec{b} = - \hat{n} \times \vec{b}$ $∴$ $\vec{c} \times (\vec{a} \times \vec{b}) = \vec{c} \times (- \hat{n} \times \vec{b})$

$= (\vec{c} \cdot \vec{b}) (- \hat{n}) - {\vec{c} \cdot (- \hat{n})} \vec{b}$ $= - 12 \hat{n} + (0) \vec{b} = - 12 \hat{n}$

$∴$ $| \vec{c} \times (\vec{a} \times \vec{b}) |$ $= 12$

Solution

Q.
If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = α \vec{b} - \hat{n}, (α \neq 0)$ and $\vec{b} \cdot \vec{c} = 12,$ then $| \vec{c} \times (\vec{a} \times \vec{b}) |$ is equal to [2023]

1 9

2 15

3 6

4 12

Want Us to Email you About Special Offers & Updates?

Solution

Q. If a→,b→,c→ are three non-zero vectors and n^ is a unit vector perpendicular to c→ such that a→=αb→-n^,(α≠0) and b→·c→=12, then |c→×(a→×b→)| is equal to [2023]

1 9

2 15

3 6

4 12

Ans. (4) We have, |n^|=1 Given, n^⊥c→, a→=αb→-n^ and b→·c→=12 Now, a→×b→=(αb→-n^)×b→=- n^×b→ ∴ c→×(a→×b→)=c→×(-n^×b→) =(c→·b→)(-n^)-{c→·(-n^)}b→=-12n^+(0)b→=-12n^ ∴ |c→×(a→×b→)|=12

Want Us to Email you About Special Offers & Updates?

Q.
If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = α \vec{b} - \hat{n}, (α \neq 0)$ and $\vec{b} \cdot \vec{c} = 12,$ then $| \vec{c} \times (\vec{a} \times \vec{b}) |$ is equal to [2023]