Physics

A vector can be multiplied by a scalar, which can be dimensionless or dimensional, and can also be negative. Therefore, all options A, B, and C are correct.

When the coil of a galvanometer is in equilibrium, the deflecting couple caused by the current is exactly balanced by the restoring couple produced by the spring or suspension. This balance of torques results in no net rotation, maintaining equilibrium.

The cross product of two vectors results in a vector perpendicular to both, with direction given by the right-hand rule. The cross product is anti-commutative, meaning $\mathbf{a}\times \mathbf{b}=-\mathbf{b}\times \mathbf{a}$. This implies that reversing the order of the vectors reverses the direction (sign) of the cross product vector. Rotating one vector through 270° relative to the other is equivalent to rotating it -90°, which reverses the direction of the cross product vector, effectively producing a 'negative' vector. When vectors are parallel or anti-parallel, the sine of the angle between them is zero, so the cross product is the zero vector, not a negative vector. When vectors are perpendicular, the cross product vector has maximum magnitude and a definite direction. Therefore, the cross product is a negative vector when the vectors are rotated through 270°, i.e., when the order or orientation causes the direction to reverse.

An alternating current (AC) not only changes its direction repeatedly but also varies its magnitude in a sinusoidal (harmonic) manner over time. Option D accurately describes both these characteristics, making it the most complete and correct answer.

The magnitude of the vector $\mathbf{A}=2\hat{i}-2\hat{j}+\hat{k}$ is calculated as $\sqrt{2^2+(-2{)}^2+1^2}=\sqrt{4+4+1}=\sqrt{9}=3$. The unit vector in the direction of $\mathbf{A}$ is the vector divided by its magnitude, which is $\frac{2}{3}\hat{i}-\frac{2}{3}\hat{j}+\frac{1}{3}\hat{k}$. This corresponds to option C, $\frac{2\hat{i}-2\hat{j}+\hat{k}}{3}$. Therefore, the correct answer is C.

The dot product of two vectors A and B is calculated as the sum of the products of their corresponding components: AxBx + AyBy + AzBz. Option A correctly represents this formula.

Vector addition obeys the commutative law (order doesn't matter), associative law (grouping doesn't matter), and distributive law (scalar multiplication distributes over vector addition). Therefore, all given laws apply to vector addition.

The maximum number of components a vector can have depends on the dimension of the vector space it belongs to. In an n-dimensional vector space, a vector has exactly n components. However, vector spaces can be infinite-dimensional, meaning vectors can have infinitely many components. Therefore, the maximum number of components a vector can have is infinite, not limited to 3 as in three-dimensional space.

A unit vector denoted as n^ typically represents a normal vector to a surface, meaning it is perpendicular to that surface. This is a standard notation in physics and vector mathematics.

A magnetic field exerts a force on a charged particle only when the particle is moving and its velocity has a component perpendicular to the magnetic field lines. If the particle moves parallel to the field lines, no magnetic force acts on it. Therefore, option D is correct.