Question
Example 1.1 Find the resultant of two vectors of 3 units and 4 units acting at a point O at an angle of 45° with each other. Using cosine and sine rule
Answers
GPT 3.5
To find the resultant of two vectors with given magnitudes and angles between them, we can use the cosine and sine rule.
Let's denote the two vectors as A and B, with magnitudes A = 3 units and B = 4 units. The angle between them is given as 45°.
Using the cosine rule, we can find the magnitude of the resultant vector R:
R^2 = A^2 + B^2 - 2ABcos(45°)
R^2 = 3^2 + 4^2 - 2(3)(4)cos(45°)
R^2 = 9 + 16 - 24cos(45°)
R^2 = 25 - 24cos(45°)
R^2 ≈ 25 - 24(0.7071) [using the value of cos(45°) = 0.7071 from the table]
R^2 ≈ 25 - 16.97184
R^2 ≈ 8.02816
R ≈ √8.02816
R ≈ 2.834 units
So, the magnitude of the resultant vector R is approximately 2.834 units.
Now, let's use the sine rule to find the angle that R makes with vector A.
sin(θ) / A = sin(45°) / R
sin(θ) = (A/R) * sin(45°)
sin(θ) = (3 / 2.834) * 0.7071 [using the calculated value of R]
sin(θ) ≈ 0.7559
θ ≈ sin^(-1)(0.7559)
θ ≈ 49.38°
Therefore, the angle that the resultant vector R makes with vector A is approximately 49.38°.
Let's denote the two vectors as A and B, with magnitudes A = 3 units and B = 4 units. The angle between them is given as 45°.
Using the cosine rule, we can find the magnitude of the resultant vector R:
R^2 = A^2 + B^2 - 2ABcos(45°)
R^2 = 3^2 + 4^2 - 2(3)(4)cos(45°)
R^2 = 9 + 16 - 24cos(45°)
R^2 = 25 - 24cos(45°)
R^2 ≈ 25 - 24(0.7071) [using the value of cos(45°) = 0.7071 from the table]
R^2 ≈ 25 - 16.97184
R^2 ≈ 8.02816
R ≈ √8.02816
R ≈ 2.834 units
So, the magnitude of the resultant vector R is approximately 2.834 units.
Now, let's use the sine rule to find the angle that R makes with vector A.
sin(θ) / A = sin(45°) / R
sin(θ) = (A/R) * sin(45°)
sin(θ) = (3 / 2.834) * 0.7071 [using the calculated value of R]
sin(θ) ≈ 0.7559
θ ≈ sin^(-1)(0.7559)
θ ≈ 49.38°
Therefore, the angle that the resultant vector R makes with vector A is approximately 49.38°.