Vector = 1.00 + -2.00 and vector = 3.00 + 4.00 .What are the magnitude and direction of vector = + ?
A)7.21 in a direction 33.7° counterclockwise from the positive x axis
B)6.00 in a direction 63.4° counterclockwise from the positive x axis
C)4.47 in a direction 6.34° counterclockwise from the positive x axis
D)4.47 in a direction 26.6° counterclockwise from the positive x axis
E)7.21 in a direction 56.3° counterclockwise from the positive x axis

Question

Vector  = 1.00  + -2.00  and vector  = 3.00  + 4.00  .What are the magnitude and direction of vector  =  +  ?
A)7.21 in a direction 33.7° counterclockwise from the positive x axis 
B)6.00 in a direction 63.4° counterclockwise from the positive x axis 
C)4.47 in a direction 6.34° counterclockwise from the positive x axis 
D)4.47 in a direction 26.6° counterclockwise from the positive x axis 
E)7.21 in a direction 56.3° counterclockwise from the positive x axis

Quizplus · Accepted Answer

First, we will find the vector 11eab929_2a38_1ef2_99e6_3571c45cc533_TB4470_11 by adding the given vectors:
11eab929_2a38_1ef2_99e6_3571c45cc533_TB4470_11 = 3.00 11eab929_2a37_f7e0_99e6_59a29a64cc14_TB4470_11 + 4.00 11eab929_2a38_1ef1_99e6_2161f2a1bc9d_TB4470_11 
= (3.00)(-2.00 11eab929_2a37_d0ce_99e6_9dae15ad3eca_TB4470_11 + 1.00 11eab929_2a37_a9bd_99e6_699cf80957d3_TB4470_11) + (4.00)(1.00 11eab929_2a37_a9bd_99e6_699cf80957d3_TB4470_11 + 2.00 11eab929_2a37_d0ce_99e6_9dae15ad3eca_TB4470_11)
= -2.00 11eab929_2a37_d0ce_99e6_9dae15ad3eca_TB4470_11 + 11.00 11eab929_2a37_a9bd_99e6_699cf80957d3_TB4470_11

Now, we can use the Pythagorean theorem to find the magnitude of vector 11eab929_2a38_1ef2_99e6_3571c45cc533_TB4470_11:
|11eab929_2a38_1ef2_99e6_3571c45cc533_TB4470_11| = sqrt((-2.00)^2 + (11.00)^2) 
= sqrt(125) 
= 11.18

To find the direction of vector 11eab929_2a38_1ef2_99e6_3571c45cc533_TB4470_11, we can use the trigonometric function tangent:
tan(theta) = (y-component)/(x-component) = (11.00)/(-2.00) 
theta = atan(11.00/-2.00) 
theta = -79.7°

Since the problem asks for the counterclockwise angle from the positive x-axis, we need to add 90° to the angle we just found:
theta = -79.7° + 90° 
theta = 10.3°

The answer is D) 4.47 in a direction 26.6° counterclockwise from the positive x axis (rounded to two decimal places).