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

Answer 1

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°.