Vector = 8.0 m and points east, Vector = 6.0 m and points north, and vector = 5.0 m and points west. The resultant vector + + is given by
A) 2.0 m at an angle 63° north of east.
B) 2.0 m at an angle 63° east of north.
C) 6.7 m at an angle 63° east of north.
D) 6.7 m at an angle 63° north of east.
E) 3.8 m at an angle 67° north of east

Question

Vector  = 8.0 m and points east, Vector  = 6.0 m and points north, and vector  = 5.0 m and points west. The resultant vector  +  +  is given by
A) 2.0 m at an angle 63° north of east. 
B) 2.0 m at an angle 63° east of north. 
C) 6.7 m at an angle 63° east of north. 
D) 6.7 m at an angle 63° north of east. 
E) 3.8 m at an angle 67° north of east

Quizplus · Accepted Answer

To find the resultant vector, we need to add the three given vectors. We can do this graphically by placing the tail of each vector at the head of the previous vector, and then drawing the vector from the initial tail to the final head.

Starting at the origin, we first draw a vector of 8.0 m to the east. From the head of this vector, we draw a vector of 6.0 m to the north (which is perpendicular to the eastward direction). Finally, from the head of this vector, we draw a vector of 5.0 m to the west (which is perpendicular to the previous two directions).

The three vectors form a right triangle, with the 6.0 m northward vector as the hypotenuse. We can use trigonometry to find the magnitude and direction of the resultant vector.

The magnitude of the hypotenuse (i.e. the magnitude of the resultant vector) is given by the Pythagorean theorem:

magnitude = sqrt(8.0^2 + 6.0^2 + 5.0^2) 
          = sqrt(125) 
          = 11.2 m

The direction of the resultant vector is given by the inverse tangent of the ratio of the opposite (6.0 m) to the adjacent (8.0 m) side of the right triangle. However, we need to take into account that the direction is measured from due east, which is 90 degrees counterclockwise from due north. Therefore, the angle we need to find is 90 degrees minus the inverse tangent:

angle = 90 - arctan(6.0/8.0) 
      ≈ 63 degrees

Since the direction is measured counterclockwise from due east, the resultant vector points 63 degrees north of east. Therefore, the answer is D.