Vectors

Step 1: Vector Components.
The vector represents a 3-dimensional vector with components along the , , and directions. The angle made by the vector with the y-axis is the angle between the vector and the y-axis unit vector .
Step 2: Using the dot product formula.
The cosine of the angle between the vector and the y-axis is given by: Since is a unit vector, . The dot product is simply the coefficient of , which is 2. The magnitude of is: Thus, the cosine of the angle is: So, the angle is .
Step 3: Conclusion.
The correct answer is , which is option (D).

Step 1: Potential energy in SHM.
Potential energy is proportional to the square of displacement:
Step 2: Expressing energies.

Step 3: Energy at displacement .

Step 4: Conclusion.
The correct expression is .

Step 1: Using the condition for perpendicular vectors.
For two vectors and to be perpendicular, their dot product must be zero: Step 2: Finding the dot product.
The dot product of and is: Step 3: Solving for .
Simplifying: Thus, the correct answer is (A) 0.5.

Step 1: Argument of sine function.
The argument of a trigonometric function must be dimensionless.
Step 2: Dimensions of .

Step 3: Role of constant .
Since must have same dimensions,

Step 4: Conclusion.

Step 1: Condition for balanced Wheatstone bridge.
For a balanced Wheatstone bridge,

Step 2: Substituting values from the circuit.
From the given figure,

Step 3: Solving for .

Step 4: Conclusion.
The value of resistance is .

Step 1: Balance condition for meter bridge.

Step 2: Second balance condition.
When resistance becomes ,
Step 3: Solving the equations.
From both equations, solving gives
Step 4: Conclusion.
The unknown resistance is .

Step 1: Surface energy of a spherical drop.
Surface energy of a drop is:

Step 2: Number of small drops.
Let the number of drops be .
Using volume conservation:

Step 3: Initial surface energy.

Step 4: Final surface energy.

Step 5: Energy released.
Using :

Step 6: Conclusion.
Energy is released.

Step 1: Use the definition of .

Step 2: Substitute given value.

Step 3: Identify type of gas.
For a rigid diatomic gas:

Step 4: Verify ratio.

Step 5: Conclusion.
The gas is rigid diatomic with and .

Step 1: Definition of a Unit Vector.
A unit vector is a vector that has a magnitude of 1. To verify if a given vector is a unit vector, we compute its magnitude and check if it equals 1. The magnitude of the vector is: Thus, the vector is a unit vector. Step 2: Final Answer.
Thus, the unit vector is .

Step 1: Vector Components.
The vector has components . The vector is along the negative X-axis, so its components are . To find , subtract the components: Thus, the resulting vector is .
Step 2: Final Answer.
Thus, .

Step 1: Understanding the vector product.
The vector product of two vectors and is zero when the vectors are parallel. That is, and must be parallel.
Step 2: Analyzing the given vectors.
The vector is a unit vector along the positive x-axis, so . The vector must have components in both and -directions to be parallel to , and its magnitude must be 4. The correct vector that satisfies this condition is: Step 3: Conclusion.
Thus, is , corresponding to option (C).

Step 1: Using the law of cosines.
The magnitudes of the vectors and are given, so we can use the law of cosines. For , we have: where is the angle between the vectors. Since , this simplifies to: Similarly, for , we have: Step 2: Solving for .
From the two equations, we solve for and find: Step 3: Conclusion.
Thus, the angle between and is , corresponding to option (A).

Step 1: Use parallelogram law of forces.
If resultant makes equal angles with and , then .
Step 2: Use relation for resultant of two equal forces.
Step 3: Substitute given values.
Step 4: Simplify.

Step 1: Use vector identity.
Step 2: Compare with given condition.
Given: Step 3: Substitute and simplify.
Step 4: Find angle.

Step 1: Interpret dot product conditions.
Given and , hence is perpendicular to both and .

Step 2: Direction perpendicular to both vectors.
The vector perpendicular to both and is given by their cross product .

Step 3: Conclusion.
Therefore, is parallel to .