Physics clicker question: vector direction, from given components

Physics 205A, Spring Semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 3.21 (extended)

Students were asked the following clicker question (Classroom Performance System, einstruction.com) at the start of their learning cycle:

The x- and y-components of a vector are given: A_x = +1.4 m, A_y = –5.7 m. What is the direction of this vector, as measured counterclockwise from the +x axis?
(A) +76°.
(B) +104°.
(C) +256°.
(D) +284°.
(E) (I'm lost, and don't know how to answer this.)

Sections 70854, 70855
(A) : 3 students
(B) : 2 students
(C) : 5 students
(D) : 19 students
(E) : 11 students

This question was asked again after displaying the tallied results with the lack of consensus, with the following results. No comments were made by the instructor, in order to see if students were going to be able to discuss and determine the correct answer among themselves.

Sections 70854, 70855
(A) : 0 students
(B) : 5 students
(C) : 2 students
(D) : 35 students
(E) : 1 students

Correct answer: (D)

The acute angle made by this vector with the x-axis is given by:

Arctan(A_y/A_x) = 76°,

which is not placed in the proper quadrant by most calculators; this must be done by inspection. From the fact that A_x is positive and A_y is negative, this vector is in the fourth quadrant, and thus the proper angle is 360° - 76° = +256, as measured counterclockwise with respect to the positive x-axis, as is done when using the unit circle in trigonometry.

Response (A) is the raw answer from Arctan(A_y/A_x), response (B) is 180° - 76°, while response (C) is 180° + 76°.

Note that from inspection, response (D) is the only vector angle that is in the fourth quadrant, and thus can be deduced as the correct angle without explicitly doing any calculations.

Pre- to post- peer-interaction gains:
pre-interaction correct =48%
post-interaction correct = 81%
Hake, or normalized gain = 65%