If hat{i}+2 hat{j}+3 hat{k} and 3 hat{i}+2 ha | vedclass

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

Question

(A) Let the sides of the parallelogram be represented by vectors $\vec{a} = \hat{i}+2 \hat{j}+3 \hat{k}$ and $\vec{b} = 3 \hat{i}+2 \hat{j}+\hat{k}$.T

Vedclass · Accepted Answer

$\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

Vedclass · Answer

(A) Let the sides of the parallelogram be represented by vectors $\vec{a} = \hat{i}+2 \hat{j}+3 \hat{k}$ and $\vec{b} = 3 \hat{i}+2 \hat{j}+\hat{k}$.The diagonals of the parallelogram are given by $\vec{d_1} = \vec{a} + \vec{b}$ and $\vec{d_2} = \vec{a} - \vec{b}$.Calculating $\vec{d_1} = (\hat{i}+2 \hat{j}+3 \hat{k}) + (3 \hat{i}+2 \hat{j}+\hat{k}) = 4\hat{i} + 4\hat{j} + 4\hat{k} = 4(\hat{i} + \hat{j} + \hat{k})$.The unit vector parallel to $\vec{d_1}$ is $\frac{\vec{d_1}}{|\vec{d_1}|} = \frac{4(\hat{i} + \hat{j} + \hat{k})}{\sqrt{4^2 + 4^2 + 4^2}} = \frac{4(\hat{i} + \hat{j} + \hat{k})}{\sqrt{16+16+16}} = \frac{4(\hat{i} + \hat{j} + \hat{k})}{\sqrt{48}} = \frac{4(\hat{i} + \hat{j} + \hat{k})}{4\sqrt{3}} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$.Calculating $\vec{d_2} = (\hat{i}+2 \hat{j}+3 \hat{k}) - (3 \hat{i}+2 \hat{j}+\hat{k}) = -2\hat{i} + 0\hat{j} + 2\hat{k} = -2\hat{i} + 2\hat{k}$.The unit vector parallel to $\vec{d_2}$ is $\frac{-2\hat{i} + 2\hat{k}}{\sqrt{(-2)^2 + 2^2}} = \frac{-2\hat{i} + 2\hat{k}}{\sqrt{8}} = \frac{-2\hat{i} + 2\hat{k}}{2\sqrt{2}} = \frac{-\hat{i} + \hat{k}}{\sqrt{2}}$.Comparing with the given options, the correct unit vector is $\frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$.

If $\hat{i}+2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+2 \hat{j}+\hat{k}$ are sides of a parallelogram, then a unit vector parallel to one of the diagonals of the parallelogram is

Similar Questions

In the given potentiometer circuit,the length of the wire $AB$ is $3 \, m$ and its resistance is $R = 4.5 \, \Omega$. The length $AC$ for no deflection in the galvanometer is ................ $m$.

Two spherical black bodies have radii $r_1$ and $r_2$. Their surface temperatures are $T_1$ and $T_2$. If they radiate the same power,then $\frac{r_2}{r_1}$ is:

The protective coat of structural protein in viruses is made up of units called:

If $f_n(x) = \log \log \log \ldots \log x$ (where $\log$ is repeated $n$ times),then $\int (x f_1(x) f_2(x) \ldots f_n(x))^{-1} dx$ is equal to

$A$ solution of concentration $C \ g \ \text{equiv}/L$ has a specific resistance $R$. The equivalent conductance of the solution is

Vedclass Test Series

Exam Paper Generator

Online Exam Module