The acute angle between the line vec{r}=(hat{ | vedclass

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

Question

(B) The angle $	heta$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$ is given by $\sin 	heta = \left| \frac

Vedclass · Accepted Answer

$\sin^{-1}\left(\frac{2\sqrt{2}}{3}\right)$

Vedclass · Answer

(B) The angle $	heta$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$ is given by $\sin 	heta = \left| \frac{\vec{b} \cdot \vec{n}}{|\vec{b}| |\vec{n}|} ight|$.Here,$\vec{b} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{n} = 2\hat{i} + \hat{j} + \hat{k}$.Calculating the dot product: $\vec{b} \cdot \vec{n} = (1)(2) + (1)(1) + (1)(1) = 2 + 1 + 1 = 4$.Calculating the magnitudes: $|\vec{b}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$ and $|\vec{n}| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{6}$.Substituting these values into the formula: $\sin 	heta = \left| \frac{4}{\sqrt{3} \cdot \sqrt{6}} ight| = \frac{4}{\sqrt{18}} = \frac{4}{3\sqrt{2}} = \frac{4\sqrt{2}}{3 \cdot 2} = \frac{2\sqrt{2}}{3}$.Therefore,$	heta = \sin^{-1}\left(\frac{2\sqrt{2}}{3}ight)$.

The acute angle between the line $\vec{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k})$ and the plane $\vec{r} \cdot(2\hat{i}+\hat{j}+\hat{k})=5$ is

Similar Questions

The centroid of the triangle formed by the pair of straight lines $12x^2 - 20xy + 7y^2 = 0$ and the line $2x - 3y + 4 = 0$ is:

If we completely remove the decomposers from an ecosystem,its functioning will be adversely affected because

The events $A$ and $B$ have probabilities $0.25$ and $0.50$,respectively. The probability that both $A$ and $B$ occur simultaneously is $0.14$. Then,the probability that neither $A$ nor $B$ occurs is:

Opium is obtained from the latex of the unripe fruit of which plant?

Adsorption of a gas follows the Freundlich adsorption isotherm. $x$ is the mass of the gas adsorbed on mass $m$ of the adsorbent. The plot of $\log(\frac{x}{m})$ versus $\log p$ is shown in the given graph. $\frac{x}{m}$ is proportional to :

Vedclass Test Series

Exam Paper Generator

Online Exam Module