If a = (1, 1, 1) and c = (0, 1, -1) are two v | vedclass

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

Question

(D) Let $b = b_1 i + b_2 j + b_3 k$. Given $a \cdot b = 3$,we have $b_1 + b_2 + b_3 = 3$ ......$(i)$. Given $a 	imes b = c$,where $a = (1, 1, 1)$ and

Vedclass · Accepted Answer

$\left( \frac{5}{3}, \frac{2}{3}, \frac{2}{3} \right)$

Vedclass · Answer

(D) Let $b = b_1 i + b_2 j + b_3 k$. Given $a \cdot b = 3$,we have $b_1 + b_2 + b_3 = 3$ ......$(i)$. Given $a 	imes b = c$,where $a = (1, 1, 1)$ and $c = (0, 1, -1)$,we calculate the cross product: $a 	imes b = \begin{vmatrix} i & j & k \ 1 & 1 & 1 \ b_1 & b_2 & b_3 \end{vmatrix} = (b_3 - b_2)i + (b_1 - b_3)j + (b_2 - b_1)k$. Comparing this with $c = (0, 1, -1)$,we get: $b_3 - b_2 = 0$ ......$(ii)$ $b_1 - b_3 = 1$ ......$(iii)$ $b_2 - b_1 = -1$ ......$(iv)$ From $(ii)$,$b_2 = b_3$. Substituting into $(i)$,$b_1 + 2b_2 = 3$. From $(iii)$,$b_1 = 1 + b_3 = 1 + b_2$. Substituting $b_1$ into $b_1 + 2b_2 = 3$,we get $(1 + b_2) + 2b_2 = 3$,so $3b_2 = 2$,which means $b_2 = \frac{2}{3}$. Then $b_3 = \frac{2}{3}$ and $b_1 = 1 + \frac{2}{3} = \frac{5}{3}$. Thus,$b = \left( \frac{5}{3}, \frac{2}{3}, \frac{2}{3} ight)$.

If $a = (1, 1, 1)$ and $c = (0, 1, -1)$ are two vectors and $b$ is a vector such that $a \times b = c$ and $a \cdot b = 3$,then $b$ is equal to

Similar Questions

If $a, b$ and $c$ are unit vectors such that $a+b+c=0$ and the angle between $a$ and $b$ is $\frac{\pi}{3}$,then $|a \times b|+|b \times c|+|c \times a|=$

$A$ vector with magnitude of $3$ units,which is perpendicular to each of the vectors $\vec{a}=3 \hat{i}+\hat{j}-4 \hat{k}$ and $\vec{b}=6 \hat{i}+5 \hat{j}-2 \hat{k}$,is given by

Let $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c}=\vec{b} \times \vec{a}$ and $\vec{c} \cdot \vec{a}=0,$ then $\vec{c} \cdot \vec{b}$ is equal to

If $\vec{a}=\hat{i}+\hat{j}-\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ is a unit vector perpendicular to $\vec{a}$ and coplanar with $\vec{a}$ and $\vec{b}$,then the unit vector $\vec{d}$ perpendicular to both $\vec{a}$ and $\vec{c}$ is

If the vectors $\hat{i}-3 \hat{j}+2 \hat{k}$ and $-\hat{i}+2 \hat{j}$ represent the diagonals of a parallelogram,then its area will be

Vedclass Test Series

Exam Paper Generator

Online Exam Module