Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry1 Mark
Question
State True or False for the following: The angle between the line $\vec{\text{r}}=(5\hat{\text{i}}-\hat{\text{j}}-4\hat{\text{k}})+\lambda(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}})$ and the plane $\vec{\text{r}}(3\hat{\text{i}}-4\hat{\text{j}}-\hat{\text{k}})+5=0$ is $\sin^{-1}\Big(\frac{5}{2\sqrt{91}}\Big).$
✓

Answer

False.Solution:
We have, $\vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{n}}=3\hat{\text{i}}-4\hat{\text{j}}-\hat{\text{k}}$
Let $\theta$ is the angle between line and plane.
Then, $\sin\theta=\frac{|\vec{\text{b}}\cdot\vec{\text{n}}|}{|\vec{\text{b}}|\cdot|\vec{\text{n}}|}$
$=\bigg|\frac{(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k})}\cdot(3\hat{\text{i}}-4\hat{\text{j}}-\hat{\text{k}})}{\sqrt{6}\cdot\sqrt{26}}\bigg|$
$=\frac{|6+4-1|}{\sqrt{156}}=\frac{9}{2\sqrt{39}}$
$\therefore\theta=\sin^{-1}\frac{9}{2\sqrt{39}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "True False[1 Marks ]" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Which of the following statements are True or False.
If A is skew symmetric matrix, then $A^2$ is a symmetric matrix.
→
State True or False for the following:
Position vector of a point $\vec{\text{P}}$ is a vector whose initial point is origin.
→
State True or False for the statements of the following Exercise:
If A and B are matrices of order 3 and |A| = 5, |B| = 3, then |3AB| = 27 × 5 × 3 = 405.
→
State True or False for the statement.
The minimum value of n for which $\tan^{-1}\frac{\text{n}}{\pi}>\frac{\pi}{4},\ \text{n}\in\text{N},$ is valid is 5.
→
State True or False for the statements of the following Exercise:
The maximum value of $\begin{vmatrix}1&1&1\\1&1+\sin\theta&1\\1&1&1+\cos\theta\end{vmatrix}$ is $\frac{1}{2}.$
→
Which of the following statements are True or False.
A matrix denotes a number.
→
Which of the following statements are True or False.
Matrix addition is associative as well as commutative.
→
State True or False for the following:
If the foot of perpendicular drawn from the origin to a plane is (5, -3, -2), then the equation of plane is $\vec{\text{r}}\cdot(5\hat{\text{i}}-3\hat{\text{j}}-2\hat{\text{k}})=38.$
→
State True or False for the following:
Correct substitution for the solution of the differential equation of the type $\frac{\text{dy}}{\text{dx}}=\text{g}(\text{x, y})$ where g(x, y) is a homogeneous function of the degree zero is x = vy.
→
State True or False for the statement.
The principal value of $\sin^{-1}\Big[\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big]$ is $\frac{\pi}{3}.$
→