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

To find the magnitude and direction of vector 11ea83ea_6925_27c4_aeff_612a670d09ec_TB6394_11, we need to first determine its components in the x and y directions. Using vector addition and scalar multiplication, we can write:

11ea83ea_6925_27c4_aeff_612a670d09ec_TB6394_11 = (1.00 - 2.00) 11ea83ea_6925_4ed5_aeff_2d9bf2f5b921_TB6394_11 + (3.00 + 4.00) 11ea83ea_6925_4ed6_aeff_d178ccfff298_TB6394_11
                                                 = -1.00 11ea83ea_6925_4ed5_aeff_2d9bf2f5b921_TB6394_11 + 7.00 11ea83ea_6925_4ed6_aeff_d178ccfff298_TB6394_11

This means that the x component is -1.00 times the x component of 11ea83ea_6925_4ed5_aeff_2d9bf2f5b921_TB6394_11 plus 7.00 times the x component of 11ea83ea_6925_4ed6_aeff_d178ccfff298_TB6394_11, and similarly for the y component. Using the Pythagorean theorem, we can calculate the magnitude as:

|11ea83ea_6925_27c4_aeff_612a670d09ec_TB6394_11| = sqrt((-1.00)^2 + 7.00^2) = 4.47

To find the direction, we use the inverse tangent function and take into account the signs of the components:

θ = atan(y/x) + 180°(x < 0 and y > 0) - 180°(x < 0 and y < 0)

θ = atan(7.00/-1.00) + 180° = 256.3°

However, the question asks for the direction counterclockwise from the positive x axis, which means we need to subtract θ from 360°:

θ' = 360° - θ = 103.7°

Therefore, the answer is D) 4.47 in a direction 26.6° counterclockwise from the positive x axis.