If a=hat{i}+hat{j}+hat{k}, a cdot b=1 and a t | vedclass

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

Question

(D) Let vector $b=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$.Given $a \cdot b = b_1+b_2+b_3 = 1$  $(i)$Also, $a 	imes b = \begin{vmatrix} \hat{i} & \hat{j}

Vedclass · Accepted Answer

$\hat{i}$

Vedclass · Answer

(D) Let vector $b=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$.Given $a \cdot b = b_1+b_2+b_3 = 1$  $(i)$Also, $a 	imes b = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 1 & 1 \ b_1 & b_2 & b_3 \end{vmatrix} = (b_3-b_2) \hat{i} + (b_1-b_3) \hat{j} + (b_2-b_1) \hat{k}$.Given $a 	imes b = \hat{j}-\hat{k}$, we equate components:$b_3-b_2 = 0 \Rightarrow b_2 = b_3$$b_1-b_3 = 1 \Rightarrow b_1 = b_3+1$$b_2-b_1 = -1 \Rightarrow b_2 = b_1-1$Substituting these into equation $(i)$:$(b_3+1) + b_3 + b_3 = 1$$3b_3 + 1 = 1 \Rightarrow 3b_3 = 0 \Rightarrow b_3 = 0$.Thus, $b_2 = 0$ and $b_1 = 0+1 = 1$.Therefore, $b = 1 \hat{i} + 0 \hat{j} + 0 \hat{k} = \hat{i}$.Hence, option $D$ is correct.

If $a=\hat{i}+\hat{j}+\hat{k}$, $a \cdot b=1$ and $a \times b=\hat{j}-\hat{k}$, then $b=$

Similar Questions

$a$ and $b$ are unit vectors such that $a+2b$ is also a unit vector. If $\theta$ is the angle between $a$ and $b$,then $\sin \theta + \cos^3 \theta + \tan^5 \theta$ is equal to

If $a, b, c$ are coplanar vectors,then

If $7 \hat{i}-4 \hat{j}+5 \hat{k}$ is the position vector of the vertex $A$ of a tetrahedron $ABCD$ and $-\hat{i}+4 \hat{j}-3 \hat{k}$ is the position vector of the centroid of the triangle $BCD$,then the position vector of the centroid of the tetrahedron $ABCD$ is

If $a, b, c$ are unit vectors satisfying the relation $a+b+\sqrt{3} c=0$, then the angle between $a$ and $b$ is

Find the angle between the vectors $2\hat{i} + 3\hat{j} + \hat{k}$ and $2\hat{i} - \hat{j} - \hat{k}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module