Find the resultant of vectors angle A of magnitude 10units in the direction N40E and angle B of magnitude 8units in the direction S30E.
To find the resultant of vectors A and B, we need to add their components in the horizontal (east-west) and vertical (north-south) directions separately.
1. Let's break down vector A into its horizontal and vertical components:
- Magnitude: 10 units
- Direction: N40E
To find these components, we can use trigonometry. The northward component is given by:
- Vertical Component of A = magnitude * sin(angle)
- Vertical Component of A = 10 * sin(40°)
Similarly, the eastward component is given by:
- Horizontal Component of A = magnitude * cos(angle)
- Horizontal Component of A = 10 * cos(40°)
2. Now, let's break down vector B into its horizontal and vertical components:
- Magnitude: 8 units
- Direction: S30E
Using the same trigonometric formulas, we can calculate the components of vector B:
- Vertical Component of B = magnitude * sin(angle)
- Vertical Component of B = 8 * sin(30°)
- Horizontal Component of B = magnitude * cos(angle)
- Horizontal Component of B = 8 * cos(30°)
3. Now we have the horizontal and vertical components of both vectors:
- Vertical Component of A = 10 * sin(40°)
- Horizontal Component of A = 10 * cos(40°)
- Vertical Component of B = 8 * sin(30°)
- Horizontal Component of B = 8 * cos(30°)
4. To find the resultant vector, we need to add the horizontal components together and the vertical components together:
- Horizontal Component of Resultant = Horizontal Component of A + Horizontal Component of B
- Vertical Component of Resultant = Vertical Component of A + Vertical Component of B
5. Finally, we can calculate the magnitude and direction of the resultant vector using the Pythagorean theorem and trigonometry:
- Magnitude of Resultant = √((Horizontal Component of Resultant)² + (Vertical Component of Resultant)²)
- Direction of Resultant = atan(Vertical Component of Resultant / Horizontal Component of Resultant)
By plugging in the values calculated in step 3 into the equations in step 5, you will find the resultant vector's magnitude and direction.