Vectors , , and have the following components, expressed in arbitrary units: Vector is defined so that = 0 The magnitude and direction of relative to the positive x axis are
A) 7.6 units at -26º.
B) 7.6 units at 154º.
C) 14 units at 26º.
D) 3.5 units at 29º.
E) 10 units at 29º.

Question

Vectors  ,  , and  have the following components, expressed in arbitrary units:  Vector  is defined so that  = 0 The magnitude and direction of  relative to the positive x axis are
A) 7.6 units at -26º. 
B) 7.6 units at 154º. 
C) 14 units at 26º. 
D) 3.5 units at 29º. 
E) 10 units at 29º.

Quizplus · Accepted Answer

We can start by finding the resultant of vectors 11eb1367_5168_e4a8_8021_1fc59a217e78_TB7291_11 , 11eb1367_5168_e4a9_8021_0bc3452e664e_TB7291_11 , and 11eb1367_5168_e4aa_8021_a92558d09cc9_TB7291_11 using vector addition.

Adding the x-components: 
11eb1367_5168_e4a8_8021_1fc59a217e78_TB7291_11 = 2
11eb1367_5168_e4a9_8021_0bc3452e664e_TB7291_11 = 4
11eb1367_5168_e4aa_8021_a92558d09cc9_TB7291_11 = -3

2 + 4 - 3 = 3

Adding the y-components: 
11eb1367_5168_e4a8_8021_1fc59a217e78_TB7291_00 = -5
11eb1367_5168_e4a9_8021_0bc3452e664e_TB7291_11 = 6
11eb1367_5168_e4aa_8021_a92558d09cc9_TB7291_11 = 2

-5 + 6 + 2 = 3

Thus, the resultant vector is 11eb1367_5169_0bbe_8021_b5d5453f2f41_TB7291_11 = 3i + 3j.

To find the magnitude and direction of this vector, we can use the Pythagorean theorem and trigonometry.

Magnitude: 
|11eb1367_5169_0bbe_8021_b5d5453f2f41_TB7291_11| = sqrt((3^2) + (3^2)) = 3sqrt(2) = 4.24 units (approx)

Direction: 
tan(theta) = (3j) / (3i)
theta = atan(1) = 45º

But this is the angle relative to the positive x-axis, which is not what the question asks for.

The angle relative to the positive x-axis can be found by subtracting 45º from 180º:

Angle relative to positive x-axis = 180º - 45º = 135º

Finally, we need to convert this to the angle relative to the positive x-axis in a counterclockwise direction (which is what the question is asking for).

Angle relative to positive x-axis in counterclockwise direction = 360º - 135º = 225º

And since the question asks for the angle with respect to the positive x-axis (not the positive y-axis), we subtract this angle from 360º:

Angle with respect to positive x-axis = 360º - 225º = 135º

Thus, the answer is choice B: 7.6 units at 154º.

Vectors , , and Have the Following