Difference between revisions of "Resolve a Vector Into Components"
(importing article from wikihow) |
m (Update ref tag) |
||
| Line 3: | Line 3: | ||
== Steps == | == Steps == | ||
===Identifying Components by Graphing=== | ===Identifying Components by Graphing=== | ||
| − | #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.<ref>http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Resolution</ref> | + | #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.<ref name="rf1">http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Resolution</ref> |
#*For example, suppose you are beginning with a vector that represents a speed of {{convert|200|mph|km/h|abbr=on|round=1}} 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 {{convert|20|mph|km/h|abbr=on|sigfig=3}}. This represents a scale of {{convert|1|in|cm|adj=on|sigfig=2}} = 80 mph. | #*For example, suppose you are beginning with a vector that represents a speed of {{convert|200|mph|km/h|abbr=on|round=1}} 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 {{convert|20|mph|km/h|abbr=on|sigfig=3}}. This represents a scale of {{convert|1|in|cm|adj=on|sigfig=2}} = 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. | #*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. | ||
| − | #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.<ref | + | #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.<ref name="rf1" /> |
#*It will help to use an accurate ruler. For example, if you have chosen the scale of one square on your graph paper representing {{convert|20|mph|km/h|abbr=on|sigfig=3}}, and each square is {{convert|1/4|in|cm|adj=on|1|sigfig=2}}, then a vector of {{convert|200|mph|km/h|abbr=on|round=1}} will be a line that is 10 squares, or 2 1/2 inches, long. | #*It will help to use an accurate ruler. For example, if you have chosen the scale of one square on your graph paper representing {{convert|20|mph|km/h|abbr=on|sigfig=3}}, and each square is {{convert|1/4|in|cm|adj=on|1|sigfig=2}}, then a vector of {{convert|200|mph|km/h|abbr=on|round=1}} 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. | #*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. | #*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. | ||
| − | #Draw a right triangle, with the vector as hypotenuse. 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.<ref | + | #Draw a right triangle, with the vector as hypotenuse. 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.<ref name="rf1" /> |
#*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. | #*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. | #*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. | ||
| − | #Label the two component vectors. 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.”<ref | + | #Label the two component vectors. 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.”<ref name="rf1" /> |
#*Other samples of components might be “Up/Down” or “Left/Right.” | #*Other samples of components might be “Up/Down” or “Left/Right.” | ||
| − | #Measure the component vectors. 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 {{convert|1+1/4|in|cm|1|sigfig=2}} long, using a scale of {{convert|1|in|cm|adj=on|sigfig=2}} = 80 mph., would represent an easterly component of {{convert|100|mph|km/h|abbr=on|round=1}}.<ref | + | #Measure the component vectors. 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 {{convert|1+1/4|in|cm|1|sigfig=2}} long, using a scale of {{convert|1|in|cm|adj=on|sigfig=2}} = 80 mph., would represent an easterly component of {{convert|100|mph|km/h|abbr=on|round=1}}.<ref name="rf1" /> |
#*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 = {{convert|20|mph|km/h|abbr=on|sigfig=3}}, and you estimate that a component vector is 3 1/2 squares, then that vector represents 70 mph. | #*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 = {{convert|20|mph|km/h|abbr=on|sigfig=3}}, 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. | #*Repeat the measurement for both the horizontal and vertical component vectors, and label your results. | ||
===Calculating Components with Trigonometry=== | ===Calculating Components with Trigonometry=== | ||
| − | #Construct a rough sketch of the original vector. By relying on mathematical calculations, your graph does not need to be as neatly drawn. You do not need to determine any measurement scale. Just sketch a ray in the general direction of your vector. Label your sketched vector with its magnitude and the angle that it makes from the horizontal.<ref | + | #Construct a rough sketch of the original vector. By relying on mathematical calculations, your graph does not need to be as neatly drawn. You do not need to determine any measurement scale. Just sketch a ray in the general direction of your vector. Label your sketched vector with its magnitude and the angle that it makes from the horizontal.<ref name="rf1" /> |
#*For example, consider a rocket that is being fired upwards at a 60 degree angle, at a velocity of {{convert|1,500|m|ft|sp=us|sigfig=1}} per second. You would sketch a ray that points diagonally upward. Label its length “1500 m/s” and label its base angle “60°.” | #*For example, consider a rocket that is being fired upwards at a 60 degree angle, at a velocity of {{convert|1,500|m|ft|sp=us|sigfig=1}} per second. You would sketch a ray that points diagonally upward. Label its length “1500 m/s” and label its base angle “60°.” | ||
#*The diagram shown above indicates a force vector of 5 Newtons at an angle of 37 degrees from the horizontal. | #*The diagram shown above indicates a force vector of 5 Newtons at an angle of 37 degrees from the horizontal. | ||
| − | #Sketch and label the component vectors. Sketch a horizontal ray beginning at the base of your original vector, pointing in the same direction (left or right) as the original. This represents the horizontal component of the original vector. Sketch a vertical ray that connects the head of your horizontal vector to the head of your original angled vector. This represents the vertical component of the original vector.<ref | + | #Sketch and label the component vectors. Sketch a horizontal ray beginning at the base of your original vector, pointing in the same direction (left or right) as the original. This represents the horizontal component of the original vector. Sketch a vertical ray that connects the head of your horizontal vector to the head of your original angled vector. This represents the vertical component of the original vector.<ref name="rf1" /> |
#*The horizontal and vertical components of a vector represent a theoretical, mathematical way of breaking a force into two parts. Imagine the child's toy Etch-a-Sketch, with the separate "Vertical" and "Horizontal" drawing knobs. If you drew a line using only the "Vertical" knob and then followed with a line using only the "Horizontal" knob, you would end at the same spot as if you had turned both knobs together at exactly the same speeds. This illustrates how a horizontal and vertical force can act simultaneously on an object. | #*The horizontal and vertical components of a vector represent a theoretical, mathematical way of breaking a force into two parts. Imagine the child's toy Etch-a-Sketch, with the separate "Vertical" and "Horizontal" drawing knobs. If you drew a line using only the "Vertical" knob and then followed with a line using only the "Horizontal" knob, you would end at the same spot as if you had turned both knobs together at exactly the same speeds. This illustrates how a horizontal and vertical force can act simultaneously on an object. | ||
| − | #Use the sine function to calculate the vertical component. Because the components of a vector create a right triangle, you can use trigonometric calculations to get precise measurements of the components. Use the equation:<ref | + | #Use the sine function to calculate the vertical component. Because the components of a vector create a right triangle, you can use trigonometric calculations to get precise measurements of the components. Use the equation:<ref name="rf1" /> |
#*<math>\sin\theta=\frac{\text{vertical}}{\text{hypotenuse}}</math> | #*<math>\sin\theta=\frac{\text{vertical}}{\text{hypotenuse}}</math> | ||
#*For the missile example, you can calculate the vertical component by substituting the values that you know, and then simplifying, as follows: | #*For the missile example, you can calculate the vertical component by substituting the values that you know, and then simplifying, as follows: | ||
| Line 34: | Line 34: | ||
#*Label your result with the appropriate units. In this case, the vertical component represents an upward speed of {{convert|1,299|m|ft|sp=us|sigfig=1}} per second. | #*Label your result with the appropriate units. In this case, the vertical component represents an upward speed of {{convert|1,299|m|ft|sp=us|sigfig=1}} per second. | ||
#*The diagram above shows an alternate example, calculating the components of a force of 5 Newtons at a 37 degree angle. Using the sine function, the vertical force is calculated to be 3 Newtons. | #*The diagram above shows an alternate example, calculating the components of a force of 5 Newtons at a 37 degree angle. Using the sine function, the vertical force is calculated to be 3 Newtons. | ||
| − | #Use the cosine function to calculate the horizontal component. In the same way that you use sine to calculate the vertical component, you can use cosine to find the magnitude of the horizontal component. Use the equation:<ref | + | #Use the cosine function to calculate the horizontal component. In the same way that you use sine to calculate the vertical component, you can use cosine to find the magnitude of the horizontal component. Use the equation:<ref name="rf1" /> |
#*<math>\cos\theta=\frac{\text{horizontal}}{\text{hypotenuse}}</math> | #*<math>\cos\theta=\frac{\text{horizontal}}{\text{hypotenuse}}</math> | ||
#*Use the details from the missile example to find its horizontal component as follows: | #*Use the details from the missile example to find its horizontal component as follows: | ||
| Line 77: | Line 77: | ||
#Summarize your resultant vector. To report the resultant vector, give both its angle and magnitude. In the golf ball example, the resultant vector has a magnitude of {{convert|137.83|mph|km/h|abbr=on|sigfig=5}}, at an angle of 27.32 degrees above the horizontal. | #Summarize your resultant vector. To report the resultant vector, give both its angle and magnitude. In the golf ball example, the resultant vector has a magnitude of {{convert|137.83|mph|km/h|abbr=on|sigfig=5}}, at an angle of 27.32 degrees above the horizontal. | ||
===Reviewing Vectors and Their Components=== | ===Reviewing Vectors and Their Components=== | ||
| − | #Recall the definition of a vector. A vector is a mathematical tool that is used in physics to represent the way forces act on an object. A vector is said to represent two elements of the force, its direction and its magnitude.<ref>https://www.wired.com/2015/06/define-vector/</ref> | + | #Recall the definition of a vector. A vector is a mathematical tool that is used in physics to represent the way forces act on an object. A vector is said to represent two elements of the force, its direction and its magnitude.<ref name="rf2">https://www.wired.com/2015/06/define-vector/</ref> |
#*For example, for any moving object, you can describe its movement by giving the direction of its travel and its speed. You might say a plane is moving, for example, in a northwest direction at {{convert|500|mph|km/h|abbr=on|round=1}}. Northwest is the direction, and {{convert|500|mph|km/h|abbr=on|round=1}} is the magnitude. | #*For example, for any moving object, you can describe its movement by giving the direction of its travel and its speed. You might say a plane is moving, for example, in a northwest direction at {{convert|500|mph|km/h|abbr=on|round=1}}. Northwest is the direction, and {{convert|500|mph|km/h|abbr=on|round=1}} is the magnitude. | ||
#*A dog being held on a leash experiences a vector force. The leash, when held by the owner, is being pulled diagonally upward with some measure of force. The angle of the diagonal is the direction of the vector, and the strength of the force is the magnitude. | #*A dog being held on a leash experiences a vector force. The leash, when held by the owner, is being pulled diagonally upward with some measure of force. The angle of the diagonal is the direction of the vector, and the strength of the force is the magnitude. | ||
| − | #Understand the terminology of graphing vectors. When you draw a vector, either using a precisely drawn representation on graph paper or just a rough sketch, certain geometrical terms are used.<ref>http://mathworld.wolfram.com/Vector.html</ref> | + | #Understand the terminology of graphing vectors. When you draw a vector, either using a precisely drawn representation on graph paper or just a rough sketch, certain geometrical terms are used.<ref name="rf3">http://mathworld.wolfram.com/Vector.html</ref> |
#*A vector is represented graphically by a <math>\text{ray}</math>. A ray, in geometry, is a line segment that begins at one point and, theoretically, continues infinitely in some direction. A ray is drawn by marking a point, then a line segment of appropriate length, and marking an arrowhead at the opposite end of the line segment. | #*A vector is represented graphically by a <math>\text{ray}</math>. A ray, in geometry, is a line segment that begins at one point and, theoretically, continues infinitely in some direction. A ray is drawn by marking a point, then a line segment of appropriate length, and marking an arrowhead at the opposite end of the line segment. | ||
#*The <math>\text{tail}</math> of a vector is its starting point. Geometrically, this is the endpoint of the ray. | #*The <math>\text{tail}</math> of a vector is its starting point. Geometrically, this is the endpoint of the ray. | ||
#*The <math>\text{head}</math> of a vector is the position of the arrowhead. The one key difference between a geometric ray and a vector is that, in geometry, the arrowhead of the ray represents theoretical travel of infinite distance in the given direction. A vector, however, uses the arrowhead to indicate direction, but the length of the vector ends at the tip of the line segment, to measure its magnitude. In other words, if you sketch a ray in geometry, the length is irrelevant. If you draw a vector, however, the length is very important. | #*The <math>\text{head}</math> of a vector is the position of the arrowhead. The one key difference between a geometric ray and a vector is that, in geometry, the arrowhead of the ray represents theoretical travel of infinite distance in the given direction. A vector, however, uses the arrowhead to indicate direction, but the length of the vector ends at the tip of the line segment, to measure its magnitude. In other words, if you sketch a ray in geometry, the length is irrelevant. If you draw a vector, however, the length is very important. | ||
| − | #Recall some basic trigonometry. Component parts of a vector rely on the trigonometry of right triangles. Any diagonal line segment can become the hypotenuse of a right triangle by sketching a horizontal line from one end and a vertical line from the other end. When those two lines meet, you will have defined a right triangle.<ref>https://www.mathsisfun.com/sine-cosine-tangent.html</ref> | + | #Recall some basic trigonometry. Component parts of a vector rely on the trigonometry of right triangles. Any diagonal line segment can become the hypotenuse of a right triangle by sketching a horizontal line from one end and a vertical line from the other end. When those two lines meet, you will have defined a right triangle.<ref name="rf4">https://www.mathsisfun.com/sine-cosine-tangent.html</ref> |
#*The reference angle is the angle that is made by measuring from the horizontal base of the right triangle to the hypotenuse. | #*The reference angle is the angle that is made by measuring from the horizontal base of the right triangle to the hypotenuse. | ||
#*The sine of the reference angle can be determined by dividing the length of the opposite leg by the length of the hypotenuse. | #*The sine of the reference angle can be determined by dividing the length of the opposite leg by the length of the hypotenuse. | ||
