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 of the vector u + v, we first need to find the components of each vector.
For vector u with a magnitude of 15 and a direction angle of 35°, the x-component can be found using the formula: u_x = u * cos(θ) = 15 * cos(35°) ≈ 12.27 and the y-component can be found using the formula: u_y = u * sin(θ) = 15 * sin(35°) ≈ 8.57.
For vector v with a magnitude of 18 and a direction angle of 60°, the x-component can be found using the formula: v_x = v * cos(θ) = 18 * cos(60°) = 18 * 0.5 = 9 and the y-component can be found using the formula: v_y = v * sin(θ) = 18 * sin(60°) = 18 * 0.866 = 15.588.
Now, we can add the components to find the components of vector u + v.
The x-component of u + v is: (u + v)_x = u_x + v_x = 12.27 + 9 ≈ 21.27
The y-component of u + v is: (u + v)_y = u_y + v_y = 8.57 + 15.588 ≈ 24.158
To find the magnitude of u + v, we use the formula: |u + v| = sqrt((u + v)_x^2 + (u + v)_y^2) ≈ sqrt(21.27^2 + 24.158^2) ≈ sqrt(452.47 + 584.945864) ≈ sqrt(1037.415864) ≈ 32.2
To find the direction angle of u + v, we use the formula: θ = arctan((u + v)_y / (u + v)_x) ≈ arctan(24.158 / 21.27) ≈ arctan(1.13643) ≈ 49°
Therefore, the magnitude of u + v is approximately 32.2 and the direction angle is approximately 49°.
To find the magnitude and direction angle for u + v, we can use vector addition.
Step 1: Convert the direction angles to radians.
θ₁ = 35° = 35° * (π / 180°) ≈ 0.6109 radians
θ₂ = 60° = 60° * (π / 180°) ≈ 1.0472 radians
Step 2: Use the polar to rectangular conversion to represent the vectors in Cartesian coordinates.
For vector u:
uᵢ = 15 * cos(0.6109) ≈ 12.8
uⱼ = 15 * sin(0.6109) ≈ 8.9
For vector v:
vᵢ = 18 * cos(1.0472) ≈ 9
vⱼ = 18 * sin(1.0472) ≈ 15.6
Step 3: Add the rectangular components of u and v to find the sum.
(u + v)ᵢ = uᵢ + vᵢ ≈ 12.8 + 9 ≈ 21.8
(u + v)ⱼ = uⱼ + vⱼ ≈ 8.9 + 15.6 ≈ 24.5
Step 4: Find the magnitude of the sum using the Pythagorean theorem.
Magnitude of u + v = sqrt((u + v)ᵢ^2 + (u + v)ⱼ^2)
= sqrt(21.8^2 + 24.5^2)
≈ sqrt(475.24 + 600.25)
≈ sqrt(1075.49)
≈ 32.8 (rounded to the nearest tenth)
Step 5: Find the direction angle of the sum using the inverse tangent function.
Direction angle of u + v = arctan((u + v)ⱼ / (u + v)ᵢ)
= arctan(24.5 / 21.8)
≈ arctan(1.1225)
≈ 48° (rounded to the nearest whole degree)
Therefore, the magnitude of u + v is approximately 32.8 and the direction angle is approximately 48°.