Physics vector question

Question

Add the following four vectors together using the computer method. Don't forget to include magnitude, units, and direction on your final answer.

Vector A: 6 m, 30 degrees x-component y-component

Vector B: 4 m, 150 degrees

Vector C: 5 m, 210 degrees

Vector D: 7 m, 315 degrees

Answer 1

Not sure what the computer method is but the two main ways to add vectors are to draw them out to scale head to tail and find where the new end point is. Or break them down into the X and Y components as shown here:

Vector A

x-component = 6 COS(30) = 6 (0.866) = 5.196 m

y-component = 6 SIN(30) = 6 (0.5) = 3 m

Vector B

x-part = 4 COS(150) = 4 (-0.866) = -3.464 m

y-part = 4 SIN(150) = 4(0.5) = 2 m

Vector C

x-part = 5 COS(210) = 5 (-0.866) = -4.33 m

y-part = 5 SIN(210) = 5 (-.5) = -2.5m

Vector D

x-part = 7 COS(315) = 7(0.707) = 4.95 m

y-part = 7 SIN(315) = 7(-0.707) = -4.95 m

Now when you add the vectors up, you are adding up the X components together and then the Y components together.

X-parts = 5.196 + -3.46 + -4.33 + 4.95 = 2.352 m

Y-parts = 3+2+ -2.5 + -4.95 = -2.45m

The direction is given from TAN(θ) = Y/X

TAN(θ) = -2.45/2.352 = -1.041

θ = ATAN(-1.041) = -46.15 deg (Which is a fine angle but someone might want a positive angle)

to get that angle going clockwise you would subtract this from 360 deg to get 313.8 deg

The magnitude is obtained using the right triangle formula a^2 + b^2 = c^2

c = Square Root(a^2 +b^2)

= Sq Rt (2.352^2+ (-2.45)^2)

= Sq Rt (5.531 + 6.003)

= Sq Rt (11.534)

= 3.396 m

Answer 2

To break the vectors into components you need to use trig ratios. I'll do the first one for you:

Vector A:

Solving for the x-component:

cos(θ) = adjacent/hypotenuse so:

cos(30°) = x/6

6cos(30°) = x

x = 5.196 m

Solving for the y-component:

sin(θ) = opposite/hypotenuse so:

sin(30°) = y/6

6sin(30°) = y

y = 3 m

Complete the others in the same way. Not that you will need to add negative signs as applicable. For Vector B, the x-component will be negative because it is pointing in the negative x-direction. For Vector C, both the x-component and the y-component will be negative. For Vector D, the y-component will be negative. Once finished, add all the x-components together and that will give you the x-component of the sum (or resultant). Repeat for the y-components.

Finally, you must put the two components back together into the final vector. You do that in two parts:

1) Magnitude: To find the magnitude using the x-component and y-component, use the Pythagorean Theorem.

2) Direction: To find the direction, use inverse tangent.

The answer is:

Sum of the x-components: 2.35167

Sum of the y-components: -2.44975

Magnitude of the resultant vector: √[(2.35167)² + (-2.44975)²] = √11.53162 = 3.396 meters

Direction: tan^-1(-2.44975/2.35167) = -46.17° = 313.83°

The sum is the vector with magnitude and units 3.396 meters in a direction of 313.83°