Magnitude of u = 15, direction angle θ = 35°
Magnitude of v = 18, direction angle θ = 60°
Find the magnitude and direction angle for u + v.
Round the magnitude to the nearest tenth and the direction angle to the nearest whole degree.
To find the magnitude and direction angle for u + v, we can use the following formulas:
Magnitude of u + v = √((magnitude of u)^2 + (magnitude of v)^2 + 2(magnitude of u)(magnitude of v)cos(θv - θu))
Direction angle for u + v = θu + arccos((magnitude of u + magnitude of v)cos(θu - θv)/(magnitude of u + magnitude of v))
Plugging in the given values:
Magnitude of u = 15
Magnitude of v = 18
θu = 35°
θv = 60°
Magnitude of u + v = √((15)^2 + (18)^2 + 2(15)(18)cos(60°-35°))
= √(225 + 324 + 540cos(25°))
≈ √(549 + 540(0.9063))
≈ √(549 + 490.18)
≈ √1039.18
≈ 32.2 (rounded to the nearest tenth)
Direction angle for u + v = 35° + arccos((15 + 18)cos(35° - 60°)/(15 + 18))
= 35° + arccos(33cos(-25°)/33)
= 35° + arccos(cos(-25°))
≈ 35° + (-25°)
≈ 10° (rounded to the nearest whole degree)
Therefore, the magnitude of u + v is approximately 32.2 and the direction angle is approximately 10°.
To find the magnitude and direction angle for u + v, we can use vector addition.
Step 1: Resolve vector u into its x and y components:
The x-component of u can be found using cos(θ) = adjacent/hypotenuse:
ux = 15 * cos(35°)
The y-component of u can be found using sin(θ) = opposite/hypotenuse:
uy = 15 * sin(35°)
Step 2: Resolve vector v into its x and y components:
The x-component of v can be found using cos(θ) = adjacent/hypotenuse:
vx = 18 * cos(60°)
The y-component of v can be found using sin(θ) = opposite/hypotenuse:
vy = 18 * sin(60°)
Step 3: Add the x-components and y-components separately:
Resultant x-component = ux + vx
Resultant y-component = uy + vy
Step 4: Find the magnitude of the resultant using the Pythagorean theorem:
Magnitude of resultant = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
Step 5: Find the direction angle of the resultant using the arctan function:
Direction angle (θ) = arctan((Resultant y-component)/(Resultant x-component))
Let's calculate the magnitude and direction angle for u + v:
Calculating the x-components:
ux = 15 * cos(35°) = 12.22
vx = 18 * cos(60°) = 9
Calculating the y-components:
uy = 15 * sin(35°) = 8.55
vy = 18 * sin(60°) = 15.59
Adding the x-components and y-components:
Resultant x-component = ux + vx = 12.22 + 9 = 21.22
Resultant y-component = uy + vy = 8.55 + 15.59 = 24.14
Calculating the magnitude of the resultant:
Magnitude of resultant = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
= sqrt((21.22)^2 + (24.14)^2)
≈ sqrt(449.49 + 583.55)
≈ sqrt(1032.04)
≈ 32.1 (rounded to the nearest tenth)
Calculating the direction angle of the resultant:
Direction angle (θ) = arctan((Resultant y-component)/(Resultant x-component))
= arctan(24.14/21.22)
≈ arctan(1.14)
≈ 48° (rounded to the nearest whole degree)
Therefore, the magnitude of u + v is approximately 32.1 and the direction angle is approximately 48°.