ABCDEF is a regular hexagon with sides of unit length. Find the magnitude and the direction of AB+AC+AD+AE+AF.
The textbook answer: 6, 60 degrees to AB
A full solution with steps will be greatly appreciated.
Thank you :)
I put A at lower left and B a lower right.
AB = 1 i + 0 j
AC = (1+cos 60)i + sin 60 j
AD = 1 i + 2 sin 60 j
AE = 0 i + 2 sin 60 j
AF = -cos 60 i + sin 60 j
sum of x components = 3
sum of y components = 6 sin 60 = 6 (sqrt3 /2) = 3 sqrt 3
magnitude = sqrt(3^2 + 9*3) = sqrt 36 = 6 sure enough
tan angle = sqrt 3 /1
so angle = 60
https://answers.yahoo.com/question/index?qid=20090914114948AAF9qjv
Well, you're in luck! I've got a full solution for you, with a touch of humor, of course.
Now, let's break it down step by step. We want to find the magnitude and direction of the sum of AB, AC, AD, AE, and AF.
Step 1: Determine the magnitude of AB. We know that the sides of the hexagon have a unit length, so AB also has a length of 1.
Step 2: Find the direction of AB. Since ABCDEF is a regular hexagon, each angle at the center is 60 degrees. Therefore, the direction of AB is 60 degrees.
Step 3: Move on to AC. Since all sides of a regular hexagon are congruent, the magnitude of AC is also 1.
Step 4: Determine the direction of AC. We have to keep in mind that each side of the hexagon is connected to the previous side. As we move from AB to AC, we make a 60-degree turn. So, the direction of AC is also 60 degrees.
Step 5: Keep going! For AD, its magnitude is 1, just like the other sides.
Step 6: Determine the direction of AD. We've already made a 60-degree turn when we went from AB to AC. So, from AC to AD, we make another 60-degree turn. The total turn is now 120 degrees.
Step 7: Time for AE! Again, the magnitude is 1.
Step 8: Determine the direction of AE. We've already made a 120-degree turn from AB to AD. So, from AD to AE, we make another 60-degree turn. The total turn is now 180 degrees.
Step 9: Almost there! AF also has a magnitude of 1.
Step 10: Determine the direction of AF. We've already made a 180-degree turn from AB to AE. So, from AE to AF, we make another 60-degree turn. The total turn is now 240 degrees.
Step 11: Add everything up! The magnitude of the sum AB + AC + AD + AE + AF is given by adding up the magnitudes of each side. So, 1 + 1 + 1 + 1 + 1 = 5.
Step 12: Lastly, the direction of the sum is determined by the overall direction once we add up all the turns. Since we had a total of 240 degrees of turns, the direction is 240 degrees.
So, the final answer is that the magnitude of AB + AC + AD + AE + AF is 5, and the direction is 240 degrees, which is 60 degrees to AB.
I hope that put a smile on your face and helped you understand the solution. If you have any more questions, feel free to ask!
To find the magnitude and direction of AB+AC+AD+AE+AF in the regular hexagon ABCDEF, we can start by drawing a diagram and labeling the relevant points.
Step 1: Draw the regular hexagon ABCDEF with sides of unit length.
Step 2: Label the points where the sides intersect. Let the center of the hexagon be point O.
Step 3: Draw the vectors AB, AC, AD, AE, and AF, using arrows to indicate the direction and length of each vector.
Step 4: Since all sides of the hexagon have a length of 1 unit, all the vectors have the same magnitude of 1 unit.
Step 5: Note that by starting at A and adding all the vectors AB, AC, AD, AE, and AF, we will end up at point F.
Step 6: To find the magnitude of AB+AC+AD+AE+AF, we need to find the distance between points A and F. Since AB, AC, AD, AE, and AF are all unit vectors, the distance from A to F is equal to the magnitude of the sum of these vectors, which is 5 units.
Step 7: To find the direction of AB+AC+AD+AE+AF, we can draw a line from A to F and then draw a line from the origin point O perpendicular to AF. This perpendicular line represents a direction called the bearing of AF.
Step 8: Since ABCDEF is a regular hexagon, the angle between adjacent sides is 60 degrees. Therefore, the angle between AF and the horizontal line passing through the origin point O is 60 degrees.
Step 9: The direction of AB+AC+AD+AE+AF is 60 degrees to the right of AB.
Therefore, the magnitude of AB+AC+AD+AE+AF is 5 units, and it is 60 degrees to the right of AB.
To find the magnitude and direction of AB+AC+AD+AE+AF, we can break down the problem into smaller steps.
Step 1: Draw the Hexagon
Start by drawing a regular hexagon ABCDEF with sides of unit length. Label the vertices as shown:
A
/ \
F B
| |
E C
\ /
D
Step 2: Label the Vectors
Label each of the vectors AB, AC, AD, AE, and AF.
Step 3: Draw Vector Diagram
Draw a vector diagram, starting at point A, to represent each of the vectors AB, AC, AD, AE, and AF. Make sure each vector is in the correct orientation and length on the diagram.
Step 4: Analyze Vector Diagram
Now, look at the vector diagram and analyze the problem. Since all sides of the hexagon have a unit length, each vector AB, AC, AD, AE, and AF has a magnitude of 1.
Step 5: Determine the Resultant Vector
To find the magnitude and direction of the resultant vector AB+AC+AD+AE+AF, start at point A and move along the vectors in order: AB, AC, AD, AE, and AF. The magnitude of the resultant vector is the sum of the magnitudes of the individual vectors, which in this case is 1 + 1 + 1 + 1 + 1 = 5.
Step 6: Determine the Direction
To determine the direction of the resultant vector, draw a line from the initial point (A) to the final point (F). This line represents the direction of the resultant vector.
Step 7: Measure the Angle
Using a protractor or angle measuring tool, measure the angle between the resultant vector and the vector AB. In this case, the angle is 60 degrees.
So, the magnitude of AB+AC+AD+AE+AF is 5, and the direction is 60 degrees to AB.