Let vec{a}, vec{b} and vec{c} be three non-co | vedclass

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

Question

(D) Given $\vec{a}, \vec{b}, \vec{c}$ are unit vectors with angle $\frac{\pi}{3}$ between any pair,so $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} =

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(D) Given $\vec{a}, \vec{b}, \vec{c}$ are unit vectors with angle $\frac{\pi}{3}$ between any pair,so $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = \cos(\frac{\pi}{3}) = \frac{1}{2}$.Given $p \vec{a} + q \vec{b} + r \vec{c} = \vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c}$.Taking the dot product with $\vec{a}, \vec{b}, \vec{c}$ respectively:$1$) $\vec{a} \cdot (p \vec{a} + q \vec{b} + r \vec{c}) = \vec{a} \cdot (\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c}) \Rightarrow p + \frac{q}{2} + \frac{r}{2} = [\vec{a} \vec{b} \vec{c}]$.$2$) $\vec{b} \cdot (p \vec{a} + q \vec{b} + r \vec{c}) = \vec{b} \cdot (\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c}) \Rightarrow \frac{p}{2} + q + \frac{r}{2} = 0$.$3$) $\vec{c} \cdot (p \vec{a} + q \vec{b} + r \vec{c}) = \vec{c} \cdot (\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c}) \Rightarrow \frac{p}{2} + \frac{q}{2} + r = [\vec{a} \vec{b} \vec{c}]$.From $(1)$ and $(3)$,$p + \frac{q}{2} + \frac{r}{2} = \frac{p}{2} + \frac{q}{2} + r \Rightarrow p = r$.Substituting $r = p$ into $(2)$: $\frac{p}{2} + q + \frac{p}{2} = 0 \Rightarrow p + q = 0 \Rightarrow q = -p$.Now,$\frac{p^2 + 2q^2 + r^2}{q^2} = \frac{p^2 + 2(-p)^2 + p^2}{(-p)^2} = \frac{p^2 + 2p^2 + p^2}{p^2} = \frac{4p^2}{p^2} = 4$.

Similar Questions

The volume of the parallelopiped whose coterminous edges are $\hat{j}+\hat{k}$, $\hat{i}+\hat{k}$, and $\hat{i}+\hat{j}$ is

If $a = i - j + k$,$b = i + 2j - k$ and $c = 3i + pj + 5k$ are coplanar,then the value of $p$ is:

For what value of $a$ is the volume of the parallelepiped formed by the vectors $\hat{i} + a\hat{j} + \hat{k}$,$\hat{j} + a\hat{k}$,and $a\hat{i} + \hat{k}$ minimum?

For three vectors $u, v, w$,which of the following expressions is not equal to any of the remaining three?

The volume of the parallelepiped whose coterminous edges are represented by the vectors $2i - 3j + 4k$,$i + 2j - 2k$,and $3i - j + k$ is ............ $cubic \ units$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module