The unit vectors orthogonal to 3 hat{i}+2 hat | vedclass

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

Question

(B) Let the required vector be $\vec{v} = a(2 \hat{i}+\hat{j}+\hat{k}) + b(\hat{i}-\hat{j}+\hat{k})$.This simplifies to $\vec{v} = (2a+b)\hat{i} + (a-

Vedclass · Accepted Answer

$\pm \frac{1}{\sqrt{10}}(3 \hat{j}-\hat{k})$

Vedclass · Answer

(B) Let the required vector be $\vec{v} = a(2 \hat{i}+\hat{j}+\hat{k}) + b(\hat{i}-\hat{j}+\hat{k})$.This simplifies to $\vec{v} = (2a+b)\hat{i} + (a-b)\hat{j} + (a+b)\hat{k}$.Since $\vec{v}$ is orthogonal to $3 \hat{i}+2 \hat{j}+6 \hat{k}$,their dot product is zero:$3(2a+b) + 2(a-b) + 6(a+b) = 0$.$6a + 3b + 2a - 2b + 6a + 6b = 0$.$14a + 7b = 0 \implies b = -2a$.Substituting $b = -2a$ into the expression for $\vec{v}$:$\vec{v} = (2a - 2a)\hat{i} + (a - (-2a))\hat{j} + (a + (-2a))\hat{k} = 0\hat{i} + 3a\hat{j} - a\hat{k} = a(3\hat{j} - \hat{k})$.The unit vector is $\pm \frac{\vec{v}}{|\vec{v}|} = \pm \frac{a(3\hat{j} - \hat{k})}{|a|\sqrt{3^2 + (-1)^2}} = \pm \frac{3\hat{j} - \hat{k}}{\sqrt{10}}$.

The unit vectors orthogonal to $3 \hat{i}+2 \hat{j}+6 \hat{k}$ and coplanar with $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ are

Similar Questions

$a, b, c, d$ are coplanar vectors,then $(a \times b) \times (c \times d)$ is equal to

If the position vectors of the vertices of a $\triangle ABC$ are $\vec{OA} = 3\hat{i} + \hat{j} + 2\hat{k}$,$\vec{OB} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OC} = 2\hat{i} + 3\hat{j} + \hat{k}$,then the length of the altitude of $\triangle ABC$ drawn from $A$ is

Consider $P(1, 2, -3)$,$Q(-2, 1, -4)$,and $R(3, 4, -2)$. Let $\vec{B} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$,where $A_x, A_y$,and $A_z$ are the projections of the area of triangle $PQR$ on the $yz, zx$,and $xy$ planes,respectively. Then,the value of $|\vec{B}|^2$ is:

The unit vector perpendicular to both the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$ is

$A$ unit vector perpendicular to the plane of $a = 2i - 6j - 3k$ and $b = 4i + 3j - k$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module