Find the resultant of two vectors 9 units and 8 units acting at a point 0 at an angle of 45degree with each other
A. Scale drawing
B. Analytical method
To find the resultant of two vectors using a scale drawing:
1. Draw a vector representing one of the given vectors. In this case, draw a vector of 9 units in length.
2. From the tip of this vector, draw a second vector representing the second given vector. In this case, draw a vector of 8 units in length at an angle of 45 degrees from the first vector.
3. Label the initial point of both vectors as "O" and the endpoint of the second vector as "R".
4. Draw a straight line from the initial point "O" to the endpoint "R" of the second vector.
5. Measure the length of this line. This represents the magnitude of the resultant vector.
6. Draw an arrow on this line to indicate the direction of the resultant vector.
7. Label the magnitude and direction of the resultant vector.
To find the resultant of two vectors using the analytical method:
1. Represent one of the given vectors as a complex number. In this case, let the first vector be 9 units long, and we will represent it as 9 + 0i.
2. Represent the second vector as a complex number using the given angle. In this case, let the second vector be 8 units long at an angle of 45 degrees. Using trigonometry, we can represent this vector as (8 * cos(45)) + (8 * sin(45))i.
3. Add the two complex numbers together to find the resultant vector: (9 + 0i) + ((8 * cos(45)) + (8 * sin(45))i).
4. Simplify the complex number to find the magnitude and direction of the resultant vector.
To find the resultant of two vectors using both the scale drawing method and the analytical method, follow the steps outlined below:
A. Scale Drawing Method:
1. Draw a diagram representing the two vectors: One vector of 9 units in length and another vector of 8 units in length.
2. Make sure the vectors are drawn starting from the same point, which is the origin (0).
3. Draw the first vector, which is 9 units in length, at an angle of 45 degrees to the horizontal axis.
4. Next, draw the second vector, which is 8 units in length, also at an angle of 45 degrees to the horizontal axis.
5. Complete the parallelogram by drawing a line connecting the endpoints of the two vectors.
6. Measure the length of the diagonal line of the parallelogram using a ruler. This length represents the magnitude of the resultant vector.
7. The direction of the resultant can be found by measuring the angle between the horizontal axis and the diagonal line of the parallelogram.
B. Analytical Method:
1. Resolve the two vectors into their horizontal and vertical components. To do this:
- For the first vector (9 units), calculate the horizontal component by multiplying its magnitude (9) by the cosine of the angle (45 degrees).
- For the first vector (9 units), calculate the vertical component by multiplying its magnitude (9) by the sine of the angle (45 degrees).
- Repeat the above two steps to calculate the horizontal and vertical components of the second vector (8 units).
2. Add together the horizontal components of the two vectors, and add together the vertical components of the two vectors.
3. Use the Pythagorean theorem to find the magnitude of the resultant vector by calculating the square root of the sum of the squares of the horizontal and vertical components.
4. To find the angle of the resultant, use the inverse tangent function (tan^(-1)) with the ratio of the vertical component to the horizontal component.
So, to summarize:
A. Scale Drawing Method: Create a drawing representing the vectors, complete a parallelogram, and measure the resultant vector.
B. Analytical Method: Resolve the vectors into their horizontal and vertical components, add the components, find the magnitude and angle of the resultant.
A. To find the resultant of two vectors using a scale drawing, you can follow these steps:
1. Draw a coordinate system with the initial point (0, 0) as the origin.
2. Draw the first vector using a ruler, starting from the origin in the direction specified by the angle and with a length of 9 units.
3. Draw the second vector starting from the endpoint of the first vector, again using a ruler. The second vector should be in the direction specified by the angle, but with a length of 8 units.
4. Draw a straight line connecting the initial point (0, 0) to the endpoint of the second vector.
5. Measure the length of the straight line you drew in step 4 using a ruler. This represents the magnitude of the resultant vector.
B. To find the resultant of two vectors using the analytical method, you can use trigonometry and the Pythagorean theorem. Here's how:
1. Break down each vector into its x and y components. You can use trigonometry to find the x and y components of each vector.
For the first vector (9 units), the x-component is 9 cos(45°) and the y-component is 9 sin(45°).
For the second vector (8 units), the x-component is 8 cos(45°) and the y-component is 8 sin(45°).
2. Add up the x-components and the y-components separately to find the resultant x-component and the resultant y-component.
Resultant x-component = (9 cos(45°)) + (8 cos(45°))
Resultant y-component = (9 sin(45°)) + (8 sin(45°))
3. Use the Pythagorean theorem: The magnitude of the resultant vector is given by the square root of the sum of the squares of the x-component and y-component.
Resultant magnitude = √((Resultant x-component)^2 + (Resultant y-component)^2)
4. To find the direction of the resultant vector, you can use trigonometry again. The angle θ can be found using the inverse tangent function:
θ = tan^(-1)((Resultant y-component) / (Resultant x-component))
Note: Make sure to consider the appropriate quadrant for the angle.