Question:
Two adjacent sides of a parallelogram ABCD are given by \(\overline{AB} = 2\hat{i}+10\hat{j}+11\hat{k}\) and \(\overline{AD} = -\hat{i}+2\hat{j}+2\hat{k}\). The side AD is rotated by an acute angle \(α\) in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle \(α\) is given by
Two adjacent sides of a parallelogram ABCD are given by \(\overline{AB} = 2\hat{i}+10\hat{j}+11\hat{k}\) and \(\overline{AD} = -\hat{i}+2\hat{j}+2\hat{k}\). The side AD is rotated by an acute angle \(α\) in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle \(α\) is given by
Show Hint
\(AD'\) lies in the plane and is perpendicular to \(AB\); find the angle between \(AD\) and \(AB\).
Updated On: Oct 1, 2026
- \(\frac{8}{9}\)
- \(\frac{\sqrt{17}}{9}\)
- \(\frac{1}{9}\)
- \(\frac{4\sqrt{5}}{9}\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
Let \(\varphi\) be the angle between \(\vec{AB}\) and \(\vec{AD}\). The new side \(AD'\) is in the same plane, perpendicular to \(AB\), and obtained by an acute rotation.
Step 2: Key Formula or Approach
\[ \cos\varphi=\frac{\vec{AB}\cdot\vec{AD}}{|\vec{AB}||\vec{AD}|} \]
Step 3: Detailed Explanation
\(\vec{AB}\cdot\vec{AD}=-2+20+22=40\), \(|\vec{AB}|=\sqrt{4+100+121}=15\), \(|\vec{AD}|=3\).
\(\cos\varphi=\dfrac{40}{45}=\dfrac89\), so \(\sin\varphi=\dfrac{\sqrt{17}}{9}\).
To reach a direction at \(90^{\circ}\) from \(AB\), the acute rotation is \(\alpha=90^{\circ}-\varphi\).
\[ \cos\alpha=\sin\varphi=\frac{\sqrt{17}}{9} \]
Final Answer:
\(\cos\alpha=\dfrac{\sqrt{17}}{9}\), option (B). \[ \boxed{\dfrac{\sqrt{17}}{9}\ \text{(B)}} \]
Let \(\varphi\) be the angle between \(\vec{AB}\) and \(\vec{AD}\). The new side \(AD'\) is in the same plane, perpendicular to \(AB\), and obtained by an acute rotation.
Step 2: Key Formula or Approach
\[ \cos\varphi=\frac{\vec{AB}\cdot\vec{AD}}{|\vec{AB}||\vec{AD}|} \]
Step 3: Detailed Explanation
\(\vec{AB}\cdot\vec{AD}=-2+20+22=40\), \(|\vec{AB}|=\sqrt{4+100+121}=15\), \(|\vec{AD}|=3\).
\(\cos\varphi=\dfrac{40}{45}=\dfrac89\), so \(\sin\varphi=\dfrac{\sqrt{17}}{9}\).
To reach a direction at \(90^{\circ}\) from \(AB\), the acute rotation is \(\alpha=90^{\circ}-\varphi\).
\[ \cos\alpha=\sin\varphi=\frac{\sqrt{17}}{9} \]
Final Answer:
\(\cos\alpha=\dfrac{\sqrt{17}}{9}\), option (B). \[ \boxed{\dfrac{\sqrt{17}}{9}\ \text{(B)}} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Product of Two Vectors Questions
- If \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \), then the dot product \( \vec{A} \cdot \vec{B} \) is:
- If \( \mathbf{a} \) and \( \mathbf{b} \) are two non-zero vectors such that the angle between them is \( 60^\circ \), what is the probability that the dot product \( \mathbf{a} \cdot \mathbf{b} \) is positive?
- If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \), find the unit vector perpendicular to both \( \vec{a} \) and \( \vec{b} \).
- Find the angle between non-zero vectors \( \mathbf{a} \) and \( \mathbf{b} \) if their dot product \( \mathbf{a}\cdot\mathbf{b} = 0 \).
- Find the unit vector perpendicular to both \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \).
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions