Question Part Points Submissions Used Example 3.4 The Resultant Displacement A p

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Question Part
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Submissions Used
Example 3.4 The Resultant Displacement
A particle undergoes three consecutive displacements: ? 1 = (2.7 + 7.0 + 1.7) cm, ?2 = (23 - 14 - 5.0) cm, and ?3 = (-13 + 15) cm. Find the components of the resultant displacement and its magnitude.
SOLVE IT
Conceptualize Although x is sufficient to locate a point in one dimension, we need a vector to locate a point in two or three dimensions. The notation ? is a generalization of the one-dimensional displacement ?x. Three-dimensional displacements are more difficult to conceptualize than those in two dimensions because the latter can be drawn on paper.

For this problem, let us imagine that you start with your pencil at the origin of a piece of graph paper on which you have drawn x and y axes. Move your pencil 2.7 cm to the right along the x axis, then 7.0 cm upward along the y axis, and then 1.7 cm perpendicularly toward you away from the graph paper. This procedure provides the displacement described by ? 1. From this point, move your pencil 23 cm to the right parallel to the x axis, then 14 cm parallel to the graph paper in the -y direction, and then 5.0 cm perpendicularly away from you toward the graph paper. You are now at the displacement from the origin described by ? 1 + ? 2. From this point, move your pencil 13 cm to the left in the -xdirection, and (finally!) 15 cm parallel to the graph paper along the y axis. Your final position is at a displacement ? 1 + ? 2 + ? 3 from the origin.

Categorize Despite the difficulty in conceptualizing in three dimensions, we can categorize this problem as a substitution problem because of the careful bookkeeping methods that we have developed for vectors. The mathematical manipulation keeps track of this motion along the three perpendicular axes in an organized, compact way, as we see below.
To find the resultant displacement, add the three vectors:
? = ?1 + ?2 + ?3
? = (2.7 + 23 - 13) cm + (7.0 - 14 + 15) cm + (1.7 - 5.0 + 0) cm
? = (12.7 + 8 + (-3.3)) cm
Find the magnitude of the resultant vector:
R =

(Rx)2 + (Ry)2 + (Rz)2
R = cm
MASTER IT HINTS: GETTING STARTED | I'M STUCK!
An ant crawling on a table undergoes three consecutive displacements: ?1 = (4.1 + 4.6) cm, ?2 = (5.8 + 6.2) cm, and ?3 = (5.9 + 4.7) cm. Find the magnitude and direction of the resultant displacement.
magnitude cm
direction

Explanation / Answer

(1) 23.0 cm due west is 23.0 cm at 180° clockwise from the positive x-direction so vector (1) = 23.0 cos 180° i + 23.0 sin 180° j ... [using i for the horizontal component vector and j for the vertical component vector] (2)23.0 cm W 23.0° S is 23.0 cm at180 + 23 = 203° clockwise from the +x direction so vector (2) = 23.0 cos 203° i + 23.0 sin 203° j (3) E 28.0° S = 360 - 28 = 332° clockwise from the +x direction so vector (3) = 23.0 cos 332° i + 23.0 sin 332 j (4) E 35.0° N = 35.0° clockwise from the +x direction so vector (4) = 25.0 cos 35.0° i + 25.0 sin 35.0° j (a) to get the magnitude of the resultant just add all the i components to get the i component of the resultant then add the j components to get the j component of the resultant ... then use the Pythagorean theorem to get the magnitude: Adding the i components: (23.0 cos 180° + 23.0 cos 203° + 23.0 cos 332° + 25.0 cos 35.0°) i = -3.3850 i ...[the negative means that the resultant i component is in the negative direction (so points horizontally to the left)] Adding the j components: (23.0 sin 180° + 23.0 sin 203° + 23.0 sin 332 + 25.0 sin 35.0°) j = -5.4453 j ... [so the j component points in the negative direction (so vertically down)] so the resultant displacement vector is: -3.3850 i + -5.4453 j Now use the Pythagorean theorem to get the magnitude: magnitude of the resultant vector = v[(-3.3850)² + (-5.4453)²] = 6.41 cm ? EDIT ... Had to fix this ... I didn't take the v when I first did it (b) To get the direction use the tangent ratio: direction of resultant vector = arctan (-5.4453 / -3.3850) = 58.1° below the negative x direction so the direction of the resultant vector is W 58.1° S and that's equivalent to an angle of 180 + 58.1 = 238.1° clockwise from the +x direction BUT they want the direction as a positive answer with respect to due west so the direction is 58.1° south of due west

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