Question:
Let $\vec{AB}=2\hat{i}+10\hat{j}+11\hat{k}$ and $\vec{AC}=-\hat{i}+2\hat{j}+2\hat{k}$. If $\theta$ is the angle between $\vec{AB}$ and $\vec{AC}$, then $\sin\theta=$ ________.
Let $\vec{AB}=2\hat{i}+10\hat{j}+11\hat{k}$ and $\vec{AC}=-\hat{i}+2\hat{j}+2\hat{k}$. If $\theta$ is the angle between $\vec{AB}$ and $\vec{AC}$, then $\sin\theta=$ ________.
Show Hint
$\sin^2\theta + \cos^2\theta = 1$.
Updated On: Jun 26, 2026
- $\frac{\sqrt{13}}{9}$
- $\frac{\sqrt{15}}{9}$
- $\frac{\sqrt{14}}{9}$
- $\frac{\sqrt{17}}{9}$
- $\frac{4}{9}$
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Concept
Use the dot product to find $\cos\theta$, then find $\sin\theta$.
Step 2: Meaning
$\vec{AB} \cdot \vec{AC} = (2)(-1) + (10)(2) + (11)(2) = -2 + 20 + 22 = 40$.
$|\vec{AB}| = \sqrt{4+100+121} = 15$. $|\vec{AC}| = \sqrt{1+4+4} = 3$.
Step 3: Analysis
$\cos\theta = \frac{40}{15 \times 3} = \frac{40}{45} = \frac{8}{9}$.
$\sin\theta = \sqrt{1 - \cos^2\theta} = \sqrt{1 - \left(\frac{8}{9}\right)^2} = \sqrt{1 - \frac{64}{81}} = \sqrt{\frac{17}{81}}$.
Step 4: Conclusion
$\sin\theta = \frac{\sqrt{17}}{9}$. Final Answer: (D)
Use the dot product to find $\cos\theta$, then find $\sin\theta$.
Step 2: Meaning
$\vec{AB} \cdot \vec{AC} = (2)(-1) + (10)(2) + (11)(2) = -2 + 20 + 22 = 40$.
$|\vec{AB}| = \sqrt{4+100+121} = 15$. $|\vec{AC}| = \sqrt{1+4+4} = 3$.
Step 3: Analysis
$\cos\theta = \frac{40}{15 \times 3} = \frac{40}{45} = \frac{8}{9}$.
$\sin\theta = \sqrt{1 - \cos^2\theta} = \sqrt{1 - \left(\frac{8}{9}\right)^2} = \sqrt{1 - \frac{64}{81}} = \sqrt{\frac{17}{81}}$.
Step 4: Conclusion
$\sin\theta = \frac{\sqrt{17}}{9}$. Final Answer: (D)
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Product of Two Vectors Questions
- If $a , \,b$ and $c$ are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of $| a + b + c |^{2}$ is
- The value of λ for which the vectors \(2i-3j+4k\) and \(-4i+λj-8k\) are collinear is
- Let \( \vec{a} + \vec{b} = \lambda \hat{i} + 16\hat{j} - 18\hat{k} \) and \( \vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + \lambda \hat{k} \). If \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \), then \( |\vec{a}| = \)
- If \( |\vec{a}| = 12 \) and the projection of \( \vec{a} \) on \( \vec{b} \) is \( 6\sqrt{3} \), then the angle between \( \vec{a} \) and \( \vec{b} \) is
- Let \( \vec{a} = 6\hat{i} + 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is parallel to \( \vec{a} \) and \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), then \( |\vec{b}| = \)
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions