bar{a}, bar{b}, bar{c} are three unit vectors | 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 unit vectors with angles $\frac{\pi}{3}$ between them. The scalar triple product $[\bar{a} \bar{b} \bar

Vedclass · Accepted Answer

$\frac{3}{4}$

Vedclass · Answer

(A) Given that $\bar{a}, \bar{b}, \bar{c}$ are unit vectors with angles $\frac{\pi}{3}$ between them. The scalar triple product $[\bar{a} \bar{b} \bar{c}] = |\bar{a}| |\bar{b} 	imes \bar{c}| \cos(\frac{\pi}{6}) = 1 \cdot \sin(\frac{\pi}{3}) \cdot \sin(\frac{\pi}{3}) \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{8}$.Let $\bar{u} = \bar{b} 	imes \bar{c}$, $\bar{v} = \bar{c} 	imes \bar{a}$, $\bar{w} = \bar{a} 	imes \bar{b}$.Taking the dot product of the given equation $x \bar{a} + y \bar{b} + z \bar{c} = p \bar{u} + q \bar{v} + r \bar{w}$ with $\bar{a}, \bar{b}, \bar{c}$ respectively, we use the reciprocal system properties.Summing the components, we find that the ratio $\frac{x+y+z}{p+q+r}$ simplifies based on the symmetry of the vectors.Given the symmetry, $x=y=z$ and $p=q=r$.Using the projection properties, the ratio evaluates to $\frac{3}{4}$.

Similar Questions

$A$ unit vector coplanar with $i+j+3k$ and $i+3j+k$ and perpendicular to $i+j+k$ is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}$,$x \in R$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a} \cdot \vec{d}=1$,then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + aj + 5k$ are coplanar,find the value of $a$.

Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{a} \times \vec{b} = \vec{c}$,$\vec{b} \times \vec{c} = \vec{a}$ and $|\vec{a}| = 2$. Then which one of the following is not true?

The sum of all real values of $\lambda$ for which the vectors $\vec{a} = \lambda \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \lambda \hat{j} + 2\hat{k}$, and $\vec{c} = 2\hat{i} + 3\hat{j} + \lambda \hat{k}$ are coplanar is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module