Final answer:
The correct unit vector with a positive first coordinate (positive x-direction) is î (i-hat). Options â (j-hat) and ê (k-hat) have zero as their first coordinate and thus do not meet the criteria. Consequently, option (a) î (i-hat) is the correct answer.
Step-by-step explanation:
To find a unit vector in the same direction as a vector that has a positive first coordinate, we can refer to the three primary unit vectors in three-dimensional space: î (i-hat), â (j-hat), and ê (k-hat). These unit vectors represent the standard basis for the Cartesian coordinate system, indicating the positive x, y, and z directions, respectively. A unit vector is a vector that has a magnitude of 1 and points in a particular direction.
Since we are looking for a unit vector with a positive first coordinate, we need it to be in the positive direction of the x-axis. The unit vector for the positive x-direction is î (i-hat) as it has components (1, 0, 0) which clearly has a positive first coordinate for the x-axis in the rectangular coordinate system.
Choices â (j-hat) and ê (k-hat) represent the positive y and z directions and have first coordinates of 0, not positive. Therefore, the correct answer to the question is (a) î (i-hat) since it is the only vector among the options provided with a positive x-component.