12-maths Unit 2 (Vector Algebra)

Explanation
A unit vector is defined as a vector with a magnitude of 1. In this case, if v is a non-zero vector and k is a non-zero scalar, then kv is a scalar multiple of v. So, in order for kv to be a unit vector, the magnitude of kv must be 1. Since the magnitude of kv is equal to the absolute value of k times the magnitude of v, we can set the equation |k| * |v| = 1. From this equation, we can see that the only possible value for k that satisfies the equation is when |k| = 1. Therefore, the correct answer is (3).

Explanation
If a and b are two unit vectors and θ is the angle between them, then the vector a + b will not be a unit vector unless a and b are parallel or anti-parallel to each other. In all other cases, the magnitude of a + b will be greater than 1. Therefore, the correct answer is (4).

Explanation
The question is asking for the value of a certain expression involving vectors. However, the question itself is incomplete and does not provide the necessary information to determine the answer. Therefore, an explanation cannot be provided.

Explanation
The area of a parallelogram can be found by multiplying the length of one side by the perpendicular distance from that side to the opposite side. In this case, the diagonal can be considered as the base of the parallelogram, and the side can be considered as the height. Therefore, the correct answer is (4) because by using the diagonal and the side, we can find the area of the parallelogram.

Explanation
The magnitude of the vector sum of two vectors is equal to the square root of the sum of the squares of their magnitudes. Therefore, the magnitude of the vector C is equal to the square root of the sum of the squares of the magnitudes of vectors A and B. This is represented by option (2) in the given answer choices.

Explanation
The correct answer is (4) because it follows the structure of the "If...then" statement. It states that if (1) and (2) are parallel, then (3) or (4) are parallel. This means that either (3) or (4) can be parallel, so (4) is a valid answer.

Explanation
If the projection of vector a on unit vector b equals thrice the area of parallelogram P, it means that the magnitude of the projection of a on b is three times the magnitude of the area of P. This implies that the projection of a on b is three times the length of the base of P multiplied by its height. Therefore, the projection of a on b is equal to 3bh, where b is the base and h is the height of P. Hence, the correct answer is (1).

Explanation
If the projection of vector A on vector B and the projection of vector B on vector A are equal, it means that the two vectors are orthogonal or perpendicular to each other. In other words, the angle between vector A and vector B is 90 degrees. Therefore, the correct answer is (1).

Explanation
The correct answer is (4) because if a line makes angles with the positive directions of both the x and y axes, then it is in the xy-plane. The z-axis is perpendicular to the xy-plane, so the line would make a 90-degree angle with the z-axis.

Explanation
The given question presents four options, numbered 1, 2, 3, and 4, and asks for the value of an unknown variable. The correct answer is option (3).

Explanation
The vector is parallel to the line of intersection of the plane containing vector (1) and vector (2) and the plane containing vector (3) and vector (4).

Explanation
The magnitude of a vector product of two vectors is given by the product of their magnitudes multiplied by the sine of the angle between them. In this case, the magnitude of vector A is a, the magnitude of vector B is b, and the magnitude of vector C is c. The vector product of A and B is given by |A x B| = ab*sin(theta), where theta is the angle between A and B. Since A, B, and C are mutually perpendicular, the angle between A and B is 90 degrees. Therefore, sin(theta) = sin(90) = 1. Thus, |A x B| = ab*1 = ab. Similarly, |A x C| = ac and |B x C| = bc. Therefore, the value of |A x B| + |A x C| + |B x C| = ab + ac + bc, which corresponds to option (4).

Explanation
The given equation is the equation of a plane.

Explanation
If the magnitude of moment about a point of a force acting through the point is zero, it means that the force is either not acting at all or is acting in such a way that its line of action passes through the point. In other words, the force is not creating any rotational effect on the point. Therefore, the value of the magnitude of the moment is zero.

Explanation
The point of intersection of a line and a plane is the point where the line passes through the plane. In this case, the correct answer is (2) because it represents the point of intersection between the given line and plane.

Explanation
The correct answer is (4) because if two vectors are perpendicular to each other, their dot product is equal to zero. In this case, if vector a is perpendicular to vector b, then a · b = 0. Therefore, the unit vector perpendicular to both a and b can be found by taking the cross product of a and b and then normalizing it to have a magnitude of 1.

Explanation
The correct answer is (1) because the point of intersection of two lines can be determined by solving the system of equations formed by the two lines. Without any specific equations given for the lines, it is not possible to determine the exact coordinates of the point of intersection.

Explanation
The shortest distance between two parallel lines is the perpendicular distance between them. Therefore, the correct answer is (1) as it represents the perpendicular distance between the given parallel lines.

Explanation
The given lines are described as "parallel" if they do not intersect and are always the same distance apart. They are described as "intersecting" if they cross each other at a point. They are described as "skew" if they do not intersect and are not parallel. They are described as "perpendicular" if they intersect at a right angle. Therefore, since the given lines do not intersect and are not parallel, the correct answer is (3) "skew".

Explanation
The equation of a sphere is given by (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) is the center of the sphere and r is the radius.

In option (3), the center of the sphere is (3, -4, 5) and the radius is 7. Therefore, the equation of the sphere is (x - 3)^2 + (y + 4)^2 + (z - 5)^2 = 49.

This matches the given equation, so option (3) is the correct answer.