"a vector r has a magnitude of 18 units and makes a 30 degree angle with respect to the x axis. find the vector components of r using a protractor and some graph paper to verify your answer by drawing r and measuring the length of the lines representing its components." please show all steps on how to do it.
To find the vector components of a given vector, we can use trigonometry. In this case, we are given the magnitude of the vector (18 units) and the angle it makes with respect to the x-axis (30 degrees). Here are the steps:
Step 1: Draw the vector: Using a ruler and a protractor, draw a line segment representing the vector with a length of 18 units. Start this line segment at the origin (0,0) on the graph paper. The direction of the line should be at an angle of 30 degrees with respect to the x-axis.
Step 2: Draw a vertical line: From the endpoint of the vector, draw a vertical line segment that is perpendicular to the x-axis.
Step 3: Measure the length of the vertical line: Using a ruler, measure the length of the vertical line segment.
Step 4: Measure the length of the horizontal line: From the endpoint of the vector, draw a horizontal line segment that is parallel to the x-axis.
Step 5: Measure the length of the horizontal line: Using a ruler, measure the length of the horizontal line segment.
Step 6: The lengths of the vertical and horizontal lines represent the y and x components of the vector respectively.
Step 7: Verify the answer: Check if the lengths of the vertical and horizontal lines obtained from the graph correspond to the calculated values using trigonometry.
To verify the answer, use the trigonometric formulas:
- The y component of the vector can be found using: y = magnitude * sin(angle)
- The x component of the vector can be found using: x = magnitude * cos(angle)
Substituting the given values, we have:
y = 18 * sin(30) ≈ 9 units
x = 18 * cos(30) ≈ 15.588 units
Compare these calculated values with the measured lengths from the graph to verify the answer.
To find the vector components of vector 𝐫, given its magnitude and angle with respect to the x-axis, you can follow these steps using a protractor and graph paper:
Step 1: Draw a coordinate plane on the graph paper, with the x-axis and y-axis intersecting at the origin (0,0).
Step 2: Locate the point where the vector 𝐫 would end, based on its angle and magnitude. Measure 30 degrees clockwise from the positive x-axis and make a mark at a distance of 18 units from the origin. This will represent the endpoint of vector 𝐫.
Step 3: Draw a straight line from the origin to the endpoint of vector 𝐫. This line represents vector 𝐫.
Step 4: To find the x-component (𝑟𝑥) of 𝐫, drop a perpendicular line from the endpoint of vector 𝐫 to the x-axis. Measure the length of this line; this will give you the x-component of vector 𝐫.
Step 5: To find the y-component (𝑟𝑦) of 𝐫, draw a horizontal line from the endpoint of vector 𝐫 to the y-axis. Measure the length of this line; this will give you the y-component of vector 𝐫.
Step 6: To verify your answer, use a ruler to measure the lengths of the x-component and y-component lines you drew in step 4 and step 5, respectively. Make sure each measurement matches the corresponding values you have calculated.
By following these steps, you will be able to determine the vector components of 𝐫 and verify them by drawing the lines and measuring their lengths on the graph paper.