what is the resultant of two vectors of 4unit and 5unit acting on an angel 45 degree with each other ?
To find the resultant of two vectors, you can use the parallelogram law of vector addition. This law states that the resultant of two vectors can be found by drawing a parallelogram with the vectors as adjacent sides and the diagonal of the parallelogram represents the resultant.
In this case, you have two vectors of 4 units and 5 units acting at an angle of 45 degrees with each other.
First, draw the two vectors as adjacent sides of a parallelogram. At one of the vectors' starting point, draw a line at a 45-degree angle. This line represents the other vector.
Using a ruler, measure and draw the vectors. The first vector (4 units) can be represented as follows:
|
|_____________4 units
|
Now, draw the second vector (5 units) originating from the same starting point, but at a 45-degree angle:
|\
| \__________5 units
|
Now, complete the parallelogram by drawing the other two sides:
|\
| \
|____\_____
\
The diagonal of the parallelogram represents the resultant vector. Measure the length of the diagonal and that would give the magnitude of the resultant vector.
The magnitude of the resultant vector can be found using the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(C)
Where:
c is the length of the diagonal (resultant vector)
a and b are the lengths of the two vectors (4 units and 5 units)
C is the angle between the two vectors (45 degrees)
Using the formula, substitute the values:
c^2 = 4^2 + 5^2 - 2(4)(5) * cos(45 degrees)
c^2 = 16 + 25 - 40 * cos(45 degrees)
c^2 = 41 - 40 * sqrt(2)/2
c^2 ≈ 41 - 40 * 0.7071
c^2 ≈ 41 - 28.284
c^2 ≈ 12.716
c ≈ √12.716
c ≈ 3.57 units
Therefore, the magnitude of the resultant vector is approximately 3.57 units.
To find the resultant of two vectors with magnitudes of 4 units and 5 units, acting at an angle of 45 degrees with each other, we can use the parallelogram law of vector addition.
1. Draw a vector diagram: Draw the two vectors from a common point. Label one vector as A with magnitude 4 units and the other vector as B with magnitude 5 units. The angle between them is given as 45 degrees.
2. Construct a parallelogram: From the head of vector A, draw a line parallel to vector B. From the head of vector B, draw a line parallel to vector A. This forms a parallelogram.
3. Complete the parallelogram: Draw the diagonal of the parallelogram that connects the tails of the two vectors. This diagonal represents the resultant vector.
4. Measure the magnitude: Use a ruler to measure the length of the diagonal. The measured length represents the magnitude of the resultant vector.
5. Measure the direction: Use a protractor to measure the angle between the positive x-axis and the diagonal of the parallelogram. This measured angle represents the direction of the resultant vector.
6. Calculate the magnitude: Since the magnitudes of vector A and vector B are known, we can calculate the magnitude of the resultant vector using the law of cosines.
- The law of cosines states: c^2 = a^2 + b^2 - 2ab*cos(C), where c is the magnitude of the resultant vector, a and b are the magnitudes of vector A and vector B respectively, and C is the angle between them.
- In this case, a = 4, b = 5, and C = 45 degrees.
- Plugging the values into the formula: c^2 = 4^2 + 5^2 - 2*4*5*cos(45).
- Calculate: c^2 = 16 + 25 - 40*(√2)/2 = 41 - 20*(√2).
- Take the square root of both sides to find the magnitude of the resultant vector: c ≈ √(41 - 20*(√2)).
7. Calculate the direction: To find the direction of the resultant vector, we can use the law of sines.
- The law of sines states: sin(A)/a = sin(B)/b = sin(C)/c, where a, b, and c are the magnitudes of the sides of any triangle, and A, B, and C are the angles opposite to these sides.
- In this case, A is the angle between the positive x-axis and the diagonal of the parallelogram, and c is the magnitude of the resultant vector.
- We already calculated c in step 6, and we can measure A using a protractor.
- Plug the values into the formula: sin(A)/a = sin(C)/c.
- Solve for sin(A): sin(A) = a*sin(C)/c = 4*(√2)/(√(41 - 20*(√2))).
- Use the inverse sine function to find A: A ≈ arcsin(4*(√2)/(√(41 - 20*(√2)))).
8. Convert the angle to the correct quadrant: Since the given angle between the vectors is 45 degrees, and we measured A, which is the angle between the positive x-axis and the diagonal, we need to determine the correct quadrant of the resultant vector.
- If the angle between the vectors is less than 180 degrees, the resultant vector will be in the same quadrant as the positive x-axis.
- In this case, since the angle between the vectors is 45 degrees, the resultant vector will be in the same quadrant as the positive x-axis.
9. State the answer: The magnitude of the resultant vector is approximately √(41 - 20*(√2)) units, and the direction is in the same quadrant as the positive x-axis, with an angle of approximately arcsin(4*(√2)/(√(41 - 20*(√2)))) degrees.
To find the resultant of two vectors, you can use vector addition. Here's how to find the resultant of two vectors of 4 units and 5 units, acting at an angle of 45 degrees with each other:
1. Start by drawing a coordinate system with x and y axes.
2. Draw the first vector. Since it has a magnitude of 4 units, you can draw a line segment of length 4 units in any direction. Let's assume the angle between this vector and the positive x-axis is θ (theta).
3. Now, draw the second vector. Since it has a magnitude of 5 units and forms an angle of 45 degrees with the first vector, you can draw a line segment of length 5 units at an angle of 45 degrees relative to the first vector.
4. To find the resultant vector, connect the tail of the first vector to the head of the second vector. This will give you the resultant vector.
5. Measure the length of the resultant vector. This will give you the magnitude of the resultant.
6. Measure the angle between the resultant vector and the positive x-axis. This will give you the direction of the resultant vector.
By following these steps, you can determine the magnitude and direction of the resultant vector of two vectors with given magnitudes and angles. In this case, the magnitude of the resultant vector will be the length you measured in step 5, and the direction will be the angle you measured in step 6.