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
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°.
To find the resultant of two vectors, we can use the cosine rule and sine rule.
Step 1: Draw a diagram
Draw a diagram to represent the vectors. Label one vector as A with a magnitude of 3 units and another vector as B with a magnitude of 4 units. The angle between them is 45°.
Step 2: Apply the cosine rule
The cosine rule states that the square of the magnitude of the resultant vector (R) is equal to the sum of the squares of the magnitudes of the individual vectors (A and B), minus twice the product of their magnitudes and the cosine of the angle between them.
Using the cosine rule, we have:
R^2 = A^2 + B^2 - 2ABcosθ
R^2 = 3^2 + 4^2 - 2 * 3 * 4 * cos 45°
R^2 = 9 + 16 - 24 * cos 45°
Step 3: Calculate the cosine of 45°
cos 45° = (√2)/2
Substituting this value into the equation:
R^2 = 9 + 16 - 24 * (√2)/2
R^2 = 9 + 16 - 24√2
Step 4: Simplify the equation
R^2 = 25 - 24√2
Step 5: Calculate the magnitude of the resultant vector
R = √(25 - 24√2)
Step 6: Apply the sine rule
The sine rule states that the magnitude of the resultant vector (R) can be written as the sum of the magnitudes of the individual vectors (A and B), multiplied by the sine of the angle between the vectors (θ), divided by the sine of the angle formed by the resultant vector and one of the individual vectors.
Using the sine rule, we have:
R = (A * sinα)/sin(α + θ)
R = (3 * sin 45°)/sin(45° + 45°)
R = (3 * (√2)/2)/(√2/2)
Step 7: Simplify the equation
R = (3 * (√2)/2)/(√2/2)
R = (3 * √2)/1
R = 3√2 units
Therefore, the magnitude of the resultant vector is 3√2 units.