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

If $[\bar{a} \quad \bar{b} \quad \bar{c}]=4$,then the volume of the parallelepiped with coterminous edges $\bar{a}+2 \bar{b}$,$\bar{b}+2 \bar{c}$,and $\bar{c}+2 \bar{a}$ is:

If $\left[ {\vec a \,\vec b \,\vec c } \right] = 4$,then $\left[ {\vec a \times \vec b, \vec b \times \vec c, \vec c \times \vec a } \right] = \dots$

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is

Let the vectors $\vec{u} = (2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}$,$\vec{v} = (1+b) \hat{i}+2 b \hat{j}-b \hat{k}$,and $\vec{w} = (2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}$ where $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true?

If $\vec{a}, \vec{b}, \vec{c}$ are non-zero and non-coplanar vectors such that $(\vec{a} + \lambda \vec{b}) \cdot [(\vec{b} + 3\vec{c}) \times (\vec{c} - 4\vec{a})] = 0$,then $\lambda$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module