Question

The component of a vector is
A. Always less than its magnitude
B. Always greater than its magnitude
C. Always equal to its magnitude
D. None of these

Answer 1

Hint: Every vector has two different parts. Each part of the two-dimensional vector is known as a component. The components of a vector aid to describe the effect of that vector in a specific direction. The combined effect of both these components is the same as the impact of the two-dimensional single vectors.

Complete step by step solution:
The single two-dimensional vector can be substituted by the two vector components. The components of a vector in the two-dimension coordinate system are normally considered to be the \[x - \] component and the \[y - \]component. You can denote it as \[V = \left[ {\left( {v\left\{ x \right\},v\{ y\} } \right)} \right]\]. Here \[V\] is the vector. The magnitude of the component of a vector or the projection may be less than or equal to the magnitude of the vector which in turn is dependent on what we are taking as components. The magnitude of the component may be the same as the magnitude of the vector if the projection is considered along with itself, or else it will be less. For example, let us take a vector \[4\widehat i\] where \[\widehat i\] is a unit vector beside the\[x\] axis. Now the magnitude of the component of this vector along the \[x\]axis is 4, which is equal to that of the vector. Now, take a vector,\[3\widehat i + 4\widehat j\] where \[\widehat i\]and \[\widehat j\]are called unit vectors along \[x\]and \[y\]axis respectively, the magnitude of this vector is taken as \[5\] however, the magnitude of components of this vector are \[3\] and \[4\] along \[x\] and \[y\] axis respectively.Hence, Option D is correct.

Note:
The components of a vector are always direction-driven. When we take either dot product or cross product we consider the angle between the two original vectors. But while resolving a vector we must have a specific directional axis, say X-axis or Y-axis.