How to Resolve a Vector Into Components
Select an appropriate scale., Draw the vector to scale., Draw a right triangle, with the vector as hypotenuse., Label the two component vectors., Measure the component vectors.
Step-by-Step Guide
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Step 1: Select an appropriate scale.
To graph the vector and its components, you need to decide on a scale for your graph.
You need to choose a scale that is large enough to work with comfortably and accurately, but small enough that your vector can be drawn to scale.For example, suppose you are beginning with a vector that represents a speed of 200 mph (320 km/h) in a northeasterly direction.
If you are using standard graph paper of 4 squares per inch, then you might choose to have each square represent 20 mph (32.2 km/h).
This represents a scale of 1 inch (2.5 cm) = 80 mph.
There is no need to draw an x-axis and y-axis, because the vector’s placement with respect to the origin is irrelevant.
You are concerned with measuring the vector itself, and not its location in 2-dimensional or 3-dimensional space.
You are using the graph paper only as a measuring tool, so location does not matter. -
Step 2: Draw the vector to scale.
It is important that you sketch your vector as accurately as possible.
You need to represent both the correct direction and length of the vector in your drawing.It will help to use an accurate ruler.
For example, if you have chosen the scale of one square on your graph paper representing 20 mph (32.2 km/h), and each square is 1⁄4 inch (0.6 cm), then a vector of 200 mph (320 km/h) will be a line that is 10 squares, or 2 1/2 inches, long.
Use a protractor, if necessary, to show the angle or direction of the vector.
If your vector shows movement in the northeast direction, for example, you would draw a line at a 45 degree angle from the horizontal.
The direction of the vector can indicate many different kinds of direction measurements.
If you are discussing travel, it might mean a direction on the map.
To depict the path of a thrown or hit object, the angle of the vector might mean the angle of travel from the ground.
In nuclear physics, a vector might indicate the direction of an electron. , Using your ruler, begin at the tail of the vector and draw a horizontal line as wide as necessary to coincide with the head of the vector.
Mark an arrowhead at the tip of that line to indicate that this is also a component vector.
Then draw a vertical line from that point to the head of the original vector.
Mark an arrowhead at this point as well.You should have created a right triangle, consisting of three vectors.
The original vector is the hypotenuse of the right triangle.
The base of the right triangle is a horizontal vector, and the height of the right triangle is a vertical vector.
There are two exceptions, when you cannot construct a right triangle.
This will occur when the original vector is either exactly horizontal or vertical.
For a horizontal vector, the vertical component is zero, and for a vertical vector, the horizontal component is zero. , Depending on what is being represented by your original vector, you should label the two component vectors that you have just drawn.
For example, using the vector that represents travel in a northeasterly direction, the horizontal vector represents “East,” and the vertical vector represents “North.”Other samples of components might be “Up/Down” or “Left/Right.” , You can determine the magnitudes of your two component vectors using either the graph paper alone or a ruler.
If you use a ruler, then measure the length of each of the component vectors and convert using the scale you have selected.
For example, a horizontal line that is 1 1⁄4 inches (3.2 cm) long, using a scale of 1 inch (2.5 cm) = 80 mph., would represent an easterly component of 100 mph (160 km/h).If you choose to rely on the graph paper rather than a ruler, you may need to estimate a bit.
If your line crosses three full squares on the graph paper and falls in the middle of the fourth square, you would need to estimate the fraction of that last square and multiply by your scale.
For example, if 1 square = 20 mph (32.2 km/h), and you estimate that a component vector is 3 1/2 squares, then that vector represents 70 mph.
Repeat the measurement for both the horizontal and vertical component vectors, and label your results. -
Step 3: Draw a right triangle
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Step 4: with the vector as hypotenuse.
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Step 5: Label the two component vectors.
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Step 6: Measure the component vectors.
Detailed Guide
To graph the vector and its components, you need to decide on a scale for your graph.
You need to choose a scale that is large enough to work with comfortably and accurately, but small enough that your vector can be drawn to scale.For example, suppose you are beginning with a vector that represents a speed of 200 mph (320 km/h) in a northeasterly direction.
If you are using standard graph paper of 4 squares per inch, then you might choose to have each square represent 20 mph (32.2 km/h).
This represents a scale of 1 inch (2.5 cm) = 80 mph.
There is no need to draw an x-axis and y-axis, because the vector’s placement with respect to the origin is irrelevant.
You are concerned with measuring the vector itself, and not its location in 2-dimensional or 3-dimensional space.
You are using the graph paper only as a measuring tool, so location does not matter.
It is important that you sketch your vector as accurately as possible.
You need to represent both the correct direction and length of the vector in your drawing.It will help to use an accurate ruler.
For example, if you have chosen the scale of one square on your graph paper representing 20 mph (32.2 km/h), and each square is 1⁄4 inch (0.6 cm), then a vector of 200 mph (320 km/h) will be a line that is 10 squares, or 2 1/2 inches, long.
Use a protractor, if necessary, to show the angle or direction of the vector.
If your vector shows movement in the northeast direction, for example, you would draw a line at a 45 degree angle from the horizontal.
The direction of the vector can indicate many different kinds of direction measurements.
If you are discussing travel, it might mean a direction on the map.
To depict the path of a thrown or hit object, the angle of the vector might mean the angle of travel from the ground.
In nuclear physics, a vector might indicate the direction of an electron. , Using your ruler, begin at the tail of the vector and draw a horizontal line as wide as necessary to coincide with the head of the vector.
Mark an arrowhead at the tip of that line to indicate that this is also a component vector.
Then draw a vertical line from that point to the head of the original vector.
Mark an arrowhead at this point as well.You should have created a right triangle, consisting of three vectors.
The original vector is the hypotenuse of the right triangle.
The base of the right triangle is a horizontal vector, and the height of the right triangle is a vertical vector.
There are two exceptions, when you cannot construct a right triangle.
This will occur when the original vector is either exactly horizontal or vertical.
For a horizontal vector, the vertical component is zero, and for a vertical vector, the horizontal component is zero. , Depending on what is being represented by your original vector, you should label the two component vectors that you have just drawn.
For example, using the vector that represents travel in a northeasterly direction, the horizontal vector represents “East,” and the vertical vector represents “North.”Other samples of components might be “Up/Down” or “Left/Right.” , You can determine the magnitudes of your two component vectors using either the graph paper alone or a ruler.
If you use a ruler, then measure the length of each of the component vectors and convert using the scale you have selected.
For example, a horizontal line that is 1 1⁄4 inches (3.2 cm) long, using a scale of 1 inch (2.5 cm) = 80 mph., would represent an easterly component of 100 mph (160 km/h).If you choose to rely on the graph paper rather than a ruler, you may need to estimate a bit.
If your line crosses three full squares on the graph paper and falls in the middle of the fourth square, you would need to estimate the fraction of that last square and multiply by your scale.
For example, if 1 square = 20 mph (32.2 km/h), and you estimate that a component vector is 3 1/2 squares, then that vector represents 70 mph.
Repeat the measurement for both the horizontal and vertical component vectors, and label your results.
About the Author
Melissa Peterson
Professional writer focused on creating easy-to-follow practical skills tutorials.
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