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The flight paths of two aircraft are represented by the two skew lines AB and CD - Leaving Cert DCG - Question A-1 - 2010
Question A-1
The flight paths of two aircraft are represented by the two skew lines AB and CD. (a) Determine the projections of the shortest distance between the two lines. (b)... show full transcript
Worked Solution & Example Answer:The flight paths of two aircraft are represented by the two skew lines AB and CD - Leaving Cert DCG - Question A-1 - 2010
Step 1
Determine the projections of the shortest distance between the two lines.
Answer
First, create a plane containing either AB or CD and ensure it is parallel to the other line. This plane will help visualize the projection of the skew lines.
Next, draw the edge view of the plane, indicating the relevant components. The edge view provides a clearer depiction of the relationships between the skew lines and aids in visualizing the shortest distance between them.
Let X and Y be the projections of points A and B onto the plane, respectively. By projecting points from each line onto this plane, we form right angles wherever the projections meet the lines AB and CD.
Identify the shortest distance in the second auxiliary view. This involves ensuring that measurements are as accurate as possible. Use triangle properties or other geometric principles to assist in locating the desired shortest distance.
Lastly, make required projections from the edge view: project from the base to the required height while ensuring that both plan and elevation have been accurately measured.
Step 2
Determine and indicate the length of this shortest distance.
Answer
The length of the shortest distance is determined by calculating the perpendicular distance between the two skew lines. This involves using the coordinates of the projected points X and Y that you've previously identified.
Utilize the distance formula to find the length: d = rac{|(A_2 - A_1) imes (B_2 - A_1) |}{|B_2 - A_1|} Here, and are points on line AB, while and are points on line CD. Compute this value to indicate the exact shortest distance, ensuring all projections and measurements are consistent with previously established views.