Vectors

We are given that is to be resolved along the direction of , and . That implies: We compute the projection of on the direction . The required component is: Using the identity: and (because , since they are perpendicular) So,

To find the component of in the direction of , we use the formula: First, compute the dot product: Then the magnitude of is: So the component is:

To find the dot product of unit vectors and that are parallel to and , we first determine the unit vectors and then compute their dot product.
- Vector 1: . Compute its magnitude:
The unit vector parallel to is: - Vector 2: . Compute its magnitude: The unit vector parallel to is: - Dot Product: The dot product of two unit vectors is: The dot product matches option (1).
Thus, the correct answer is (1).

When two vectors are perpendicular and their resultant is half of one vector's magnitude, we can use the law of cosines or vector addition properties to determine the angle between them. After solving the equations, we find that the angle between and is . Thus, the correct answer is option (3).

The cross product of two vectors and is given by the formula: Given: Substitute into the determinant formula: Calculating the determinant: Thus, the correct answer is , and the correct option is (2).

Step 1: Understand the universal law of gravitation.
The universal law of gravitation states that the gravitational force between two masses and is directly proportional to the product of the masses and inversely proportional to the square of the distance between them: Where:
is the gravitational constant,
and are the masses of the two objects,
is the distance between the centers of the two masses.

Step 2: Convert the formula into vector form.
Since the force is a vector, the direction of the force must also be considered. The force is directed along the line joining the two masses, which is given by the unit vector . Thus, the vector form of the law is: Where is the unit vector pointing from one mass to the other. % Correct Answer Correct Answer:} (4)

Let the magnitudes of the two vectors be and , where . We are given: Since the vectors are at , we can use the Pythagorean theorem to relate the resultant magnitude: Squaring both sides: Now, using the sum of magnitudes: Substitute into the equation : Expanding: Simplifying: Dividing by 2: Solving this quadratic equation using the quadratic formula: The magnitude values for and that satisfy these conditions are and . Hence, the magnitudes of the two vectors are and .

Let and .
Dot product:
,
So,

Let , and .
Using vector addition formula:

We are given that the two vectors are of the same magnitude. Let the magnitude of each vector be , and the angle between the two vectors be . According to the problem, twice the product of the magnitudes of the two vectors is equal to the square of the magnitude of their resultant. The magnitude of the resultant can be calculated using the formula: We are told that: Substituting the expression for , we get: Simplifying: Thus, the angle between the two vectors is .

The equation implies that the vectors and are perpendicular. This can be shown by applying the Pythagorean theorem to the vector magnitudes and realizing that the sum of squares of the magnitudes of the vectors on both sides must be equal. This condition is satisfied only when the vectors are perpendicular.

Vector lies in the xy-plane. A vector perpendicular to this vector must be orthogonal, which can be any vector perpendicular to the xy-plane, i.e., along the z-axis. Hence, the vector perpendicular is:

Given each vector magnitude and angle between each two vectors . Let the vectors be , , . The resultant vector magnitude is given by: Since and ,