If bar{a}, bar{b}, bar{c} are mutually perpen | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given that $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular vectors,we have $\bar{a} \cdot \bar{b} = 0, \bar{b} \cdot \bar{c} = 0, \bar{c} \

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(A) Given that $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular vectors,we have $\bar{a} \cdot \bar{b} = 0, \bar{b} \cdot \bar{c} = 0, \bar{c} \cdot \bar{a} = 0$ and magnitudes $|\bar{a}|=1, |\bar{b}|=2, |\bar{c}|=3$. The scalar triple product is defined as $[\bar{a}+\bar{b}+\bar{c} \quad \bar{b}-\bar{a} \quad \bar{c}] = (\bar{a}+\bar{b}+\bar{c}) \cdot ((\bar{b}-\bar{a}) 	imes \bar{c})$. Expanding the cross product: $(\bar{b}-\bar{a}) 	imes \bar{c} = (\bar{b} 	imes \bar{c}) - (\bar{a} 	imes \bar{c})$. Now,the dot product becomes: $(\bar{a}+\bar{b}+\bar{c}) \cdot ((\bar{b} 	imes \bar{c}) - (\bar{a} 	imes \bar{c}))$. Since $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular,$\bar{b} 	imes \bar{c}$ is parallel to $\bar{a}$ and $\bar{a} 	imes \bar{c}$ is parallel to $\bar{b}$. Specifically,$\bar{a} \cdot (\bar{b} 	imes \bar{c}) = [\bar{a} \bar{b} \bar{c}]$ and $\bar{b} \cdot (\bar{a} 	imes \bar{c}) = -[\bar{a} \bar{b} \bar{c}]$. The expression simplifies to: $\bar{a} \cdot (\bar{b} 	imes \bar{c}) - \bar{a} \cdot (\bar{a} 	imes \bar{c}) + \bar{b} \cdot (\bar{b} 	imes \bar{c}) - \bar{b} \cdot (\bar{a} 	imes \bar{c}) + \bar{c} \cdot (\bar{b} 	imes \bar{c}) - \bar{c} \cdot (\bar{a} 	imes \bar{c})$. Terms like $\bar{a} \cdot (\bar{a} 	imes \bar{c}) = 0$ and $\bar{c} \cdot (\bar{b} 	imes \bar{c}) = 0$. Thus,we are left with $\bar{a} \cdot (\bar{b} 	imes \bar{c}) - \bar{b} \cdot (\bar{a} 	imes \bar{c}) = [\bar{a} \bar{b} \bar{c}] - (-[\bar{a} \bar{b} \bar{c}]) = 2[\bar{a} \bar{b} \bar{c}]$. Since they are mutually perpendicular,$[\bar{a} \bar{b} \bar{c}] = |\bar{a}| |\bar{b}| |\bar{c}| = 1 	imes 2 	imes 3 = 6$. Therefore,$2 	imes 6 = 12$.

If $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular vectors having magnitudes $1, 2, 3$ respectively,then $[\bar{a}+\bar{b}+\bar{c} \quad \bar{b}-\bar{a} \quad \bar{c}] = $

Similar Questions

In which of the following is the gametophytic phase dominant?

What is the chemical name of Vitamin $B_1$?

For a crystal,the angle of diffraction $(2 \theta)$ is $90^{\circ}$ and the second order line has a $d$ value of $2.28 \ \mathring{A}$. The wavelength (in $\mathring{A}$) of $X$-rays used for Bragg's diffraction is

What is the decreasing order of reactivity of $HX$ in the given reaction: $ROH + HX \rightarrow RX + H_2O$?

The minimum force required to move a body up an inclined plane is three times the minimum force required to prevent it from sliding down the plane. If the coefficient of friction between the body and the inclined plane is $\frac{1}{2 \sqrt{3}}$,then the angle of the inclined plane is (in $^{\circ}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module