Find the resultant of two vectors of 3 unit and 4 units acting at a point O at angle of 45 with each other:solve using scale drawing method and analytical method:4cm

Answer 1

To find the resultant of two vectors using the scale drawing method, follow these steps:

1. Draw a line segment to represent the first vector, measuring 3 units in length. Label one end of the line segment as point O.
2. From the endpoint of the first vector, draw a second line segment at an angle of 45 degrees to represent the second vector, measuring 4 units in length.
3. Complete a parallelogram by drawing lines parallel to the vectors from their respective endpoints.
4. Measure the length of the diagonal of the parallelogram. This represents the magnitude of the resultant vector.
5. Use a protractor to measure the angle between the resultant vector and the first vector. This represents the direction of the resultant vector.

To solve using the analytical method:

Given that the first vector is 3 units in magnitude and the second vector is 4 units in magnitude, we have vector A = 3 units and vector B = 4 units.

By using the Law of Cosines, we can find the magnitude of the resultant vector R:

R^2 = A^2 + B^2 - 2AB*cos(angle between A and B)

Substituting the given values, we have:

R^2 = 3^2 + 4^2 - 2(3)(4)*cos(45°)
R^2 = 9 + 16 - 24*cos(45°)
R^2 = 25 - 24*(sqrt(2)/2)
R^2 = 25 - 12*sqrt(2)
R^2 ≈ 3.8988

Taking the square root of both sides, we find that the magnitude of the resultant vector is approximately 1.9744.

To find the direction of the resultant vector, we can use the Law of Sines:

sin(angle between A and B) / R = sin(angle between A and resultant) / A

Substituting the given values, we have:

sin(angle between A and resultant) / 3 = sin(45°) / 1.9744

Simplifying, we have:

sin(angle between A and resultant) ≈ 0.4795

Taking the inverse sine of both sides, we find that the angle between vector A and the resultant vector is approximately 28.14°.

Therefore, the resultant vector has a magnitude of approximately 1.9744 units and is inclined at an angle of approximately 28.14° with vector A.