Question

The vector (x, y) is obtained by the following procedures: first rotate the vector clockwise by θ, then scale the vector by 3, at last, reflect it by the x-axis.

a) θ

b) 3θ

c) -θ

d) -3θ

Answer 1

The correct answer to the student's question regarding the transformations applied to a vector is option (c) -θ, corresponding to a clockwise rotation by θ, scaling, followed by reflection over the x-axis.

Step-by-step explanation:

The student's question involves the manipulation of a vector using rotations, scaling, and reflection. In vector operations, the sequence of these transformations is critical to finding the resultant vector. The process begins with a rotation which can be represented in terms of angles with respect to the positive x-axis direction. Following the rotation, the vector is scaled, which affects its magnitude but not its direction. Finally, reflection over the x-axis effectively changes the sign of the y component of the vector, leaving the x component unchanged.

To determine how the angle θ is affected by these transformations, consider the following: A clockwise rotation by θ is considered a negative rotation in standard mathematic convention. Scaling by 3 only changes the magnitude, not the direction. Reflection over the x-axis does not affect the initial rotation angle. Hence, option (c) -θ correctly represents the effect of these transformations on the initial angle of the vector.

Vector addition, rotation, and reflection are important concepts of vector manipulation that have been demonstrated in this explanation.