Find the resultant vectors of 6 units and 8 units acting at a point 0 at an angle of 45° with each other using scale drawing.
To find the resultant vector, we can use the parallelogram method in a scale drawing.
1. Draw a line segment representing the first vector of 6 units starting from point O (0) at any arbitrary direction.
2. From the endpoint of the first vector, draw a line segment representing the second vector of 8 units at an angle of 45° with respect to the first vector.
3. Complete the parallelogram by drawing two more line segments parallel to the first and second vectors.
4. The diagonal of the parallelogram connecting the initial point O (0) to the opposite corner represents the resultant vector.
5. Measure the length of the diagonal using a ruler or any measuring scale.
6. Convert the measured length back to units. Let's say the length of the diagonal is 7 units.
7. Drawing a line segment of 7 units from point 0 at the angle of 45° gives us the scale drawing of the resultant vector.
Note: The scale drawing is an approximation, and the exact numerical value of the resultant vector can be determined using appropriate mathematical calculations.
represented as complex numbers, the sum of two vectors is
6 cis0° + 8 cis45° = 12.957 cis 25.88°
To find the sum of two vectors represented as complex numbers, we simply add their real and imaginary parts separately.
The first vector is given as 6 cis 0°, which is equal to 6 + 0i.
The second vector is given as 8 cis 45°. To convert this to rectangular form, we use the formula:
r * cisθ = r * cos(θ) + i * sin(θ).
Therefore, 8 cis 45° becomes
8 * cos(45°) + i * 8 * sin(45°) = 8 * (0.7071 + 0.7071i).
Now we can add the real and imaginary parts separately:
6 + 0i + 8 * (0.7071 + 0.7071i) = 6 + 5.6568 + 5.6568i = 11.6568 + 5.6568i.
So, the sum of the two vectors is 11.6568 + 5.6568i.
To convert this back to polar form, we use the formulas:
r = √(a^2 + b^2)
θ = arctan(b/a)
where a and b are the real and imaginary parts respectively.
In this case, a = 11.6568 and b = 5.6568.
r = √(11.6568^2 + 5.6568^2) ≈ 12.957
θ = arctan(5.6568/11.6568) ≈ 25.88°
Therefore, the sum of two vectors in polar form is approximately 12.957 cis 25.88°.
To find the resultant vector of two vectors using a scale drawing, follow these steps:
Step 1: Start by drawing an accurate and scaled diagram. Choose a scale, such as 1 cm for 1 unit of length, and mark a point O as the origin at the center of your paper.
Step 2: Draw the first vector. Using a ruler and protractor, draw a line segment starting from point O that represents the first vector of 6 units. Label the endpoint of this vector as A.
Step 3: Draw the second vector. From point A, draw a line segment at an angle of 45° (using the protractor) and of length 8 units. Label the endpoint of this vector as B.
Step 4: Draw the resultant vector. Starting from point O, draw a line segment to point B. This line segment represents the resultant vector.
Step 5: Measure the length of the resultant vector using a ruler. Convert the measured length back to the original scale, where 1 cm represents 1 unit. In this case, assume the resultant vector is measured as 3 cm.
Step 6: Determine the direction of the resultant vector. Use a protractor to measure the angle between the resultant vector and a reference line, such as the positive x-axis. In this case, assume the angle is 30°.
Therefore, the resultant vectors of 6 units and 8 units acting at a point 0 at an angle of 45° with each other using a scale drawing are approximately 3 units in length and have a direction of 30°.