For non-zero vectors vec{a}, vec{b}, vec{c},t | vedclass

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

Question

(C) Given $|(\vec{a} 	imes \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$. Since $|(\vec{a} 	imes \vec{b}) \cdot \vec{c}| = |\vec{a} 	imes

Vedclass · Accepted Answer

$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$

Vedclass · Answer

(C) Given $|(\vec{a} 	imes \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$. Since $|(\vec{a} 	imes \vec{b}) \cdot \vec{c}| = |\vec{a} 	imes \vec{b}| |\vec{c}| |\cos 	heta|$,where $	heta$ is the angle between $(\vec{a} 	imes \vec{b})$ and $\vec{c}$,we have $|\vec{a} 	imes \vec{b}| |\vec{c}| |\cos 	heta| = |\vec{a}||\vec{b}||\vec{c}|$. Substituting $|\vec{a} 	imes \vec{b}| = |\vec{a}||\vec{b}| \sin \phi$,where $\phi$ is the angle between $\vec{a}$ and $\vec{b}$,we get $|\vec{a}||\vec{b}| \sin \phi |\vec{c}| |\cos 	heta| = |\vec{a}||\vec{b}||\vec{c}|$. This implies $\sin \phi = 1$ and $|\cos 	heta| = 1$. Thus,$\phi = 90^{\circ}$ (meaning $\vec{a} \perp \vec{b}$) and $	heta = 0^{\circ}$ or $180^{\circ}$ (meaning $\vec{c}$ is parallel to the normal of the plane containing $\vec{a}$ and $\vec{b}$). Since $\vec{c}$ is parallel to $\vec{a} 	imes \vec{b}$,it follows that $\vec{c} \perp \vec{a}$ and $\vec{c} \perp \vec{b}$. Therefore,$\vec{a} \cdot \vec{b} = 0$,$\vec{b} \cdot \vec{c} = 0$,and $\vec{c} \cdot \vec{a} = 0$.

For non-zero vectors $\vec{a}, \vec{b}, \vec{c}$,the condition $|(\vec{a} \times \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$ holds if and only if:

Similar Questions

The volume of the parallelepiped whose edges are represented by $-12i + \alpha k$,$3j - k$,and $2i + j - 15k$ is $546$. Then $\alpha = $

If the volume of a tetrahedron whose conterminous edges are $\overline{a}+\overline{b}, \overline{b}+\overline{c}, \overline{c}+\overline{a}$ is $24$ cubic units,then the volume of the parallelepiped whose coterminous edges are $\overline{a}, \overline{b}, \overline{c}$ is

If $35 \hat{i}+14 \hat{j}-77 \hat{k}$,$2 \hat{i}+7 \hat{j}+5 \hat{k}$ and $5 \hat{i}+2 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda$ is equal to

If $\overline{a}, \overline{b}, \overline{c}$ are mutually perpendicular vectors having magnitudes $1, 2, 3$ respectively,then the value of $[\overline{a}+\overline{b}+\overline{c} \quad \overline{b}-\overline{a} \quad \overline{c}]$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module