Question:
Vectors \(a\hat{i}+b\hat{j}+\hat{k}\) and \(2\hat{i}-3\hat{j}+4\hat{k}\) are perpendicular to each other when \(3a+2b = 7\), the ratio of a to b is \(x/2\). The value of x is
Vectors \(a\hat{i}+b\hat{j}+\hat{k}\) and \(2\hat{i}-3\hat{j}+4\hat{k}\) are perpendicular to each other when \(3a+2b = 7\), the ratio of a to b is \(x/2\). The value of x is
Show Hint
Perpendicular vectors have zero dot product; solve the two equations for a and b.
Updated On: Oct 1, 2026
- \(4\)
- \(1\)
- \(8\)
- \(3\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Use the dot product
For perpendicular vectors, \((a\hat i + b\hat j + \hat k)\cdot(2\hat i - 3\hat j + 4\hat k) = 0\), so \(2a - 3b + 4 = 0\).
Step 2: Use the given relation
We also have \(3a + 2b = 7\). From the first, \(a = \frac{3b-4}{2}\).
Step 3: Solve
\(3\cdot\frac{3b-4}{2} + 2b = 7\) gives \(9b - 12 + 4b = 14\), so \(b = 2\) and \(a = 1\).
Step 4: Find x
\(\frac ab = \frac12 = \frac x2\), so \(x = 1\). Option (B).
Final Answer:
The value of x is 1. \[ \boxed{\text{(B)}\ 1} \]
For perpendicular vectors, \((a\hat i + b\hat j + \hat k)\cdot(2\hat i - 3\hat j + 4\hat k) = 0\), so \(2a - 3b + 4 = 0\).
Step 2: Use the given relation
We also have \(3a + 2b = 7\). From the first, \(a = \frac{3b-4}{2}\).
Step 3: Solve
\(3\cdot\frac{3b-4}{2} + 2b = 7\) gives \(9b - 12 + 4b = 14\), so \(b = 2\) and \(a = 1\).
Step 4: Find x
\(\frac ab = \frac12 = \frac x2\), so \(x = 1\). Option (B).
Final Answer:
The value of x is 1. \[ \boxed{\text{(B)}\ 1} \]
Was this answer helpful?
0
0
Top MHT CET Physics Questions
- 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)
- A particle moves along a circle of radius ?$r$? with constant tangential acceleration. If the velocity of the particle is ?$?$? at the end of second revolution, after the revolution has started then the tangential acceleration is
- If $M_z$ = magnetization of a paramagnetic sample, $B$ = external magnetic field, $T$ = absolute temperature, $C$ = curie constant then according to Curie�s law in magnetism, the correct relation is
- In a fundamental mode, the time required for the sound wave to reach upto the closed end of a pipe filled with air is $'t'$ second. The frequency of vibration of air column is
- In a parallel plate capacitor, the capacity increases if
View More Questions
Top MHT CET Vectors Questions
- Approximate value of \( \cos 59^\circ \) is:
- If \( \mathbf{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}, \mathbf{b} = -\hat{i} + 2\hat{j} + \hat{k} \) and \( \mathbf{c} = 3\hat{i} + \hat{j} \) are the vectors such that \( \mathbf{a} + \lambda \mathbf{b} \) is perpendicular to \( \mathbf{c} \), then the value of \( \lambda \) is:
- If cos(12/13) + sin(P), then the value of P is:
- If the angle between the line \( 2(x + 1) = y = z \) and the plane \( 2x - y + \sqrt{2} z + 4 = 0 \) is \( \frac{\pi}{6} \), then the value of \( \lambda \) is:
- If the volume of the tetrahedron, whose vertices are \( A(1, 2, 3), B(-3, -1, 1), C(2, 1, 3) \) and \( D(-1, 2, x) \), is \( \frac{11}{6} \) cubic units, then the value of \( x \) is:
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