Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![**Solve for X**
The following diagram shows a geometric shape with several vectors labeled. The vectors are defined with magnitudes that are multiples of a variable, x. The vectors are as follows:
- Vector PQ = 4x
- Vector QP = 3x
- Vector PT = 3x
- Vector TS = 2x
- Vector SR = x
- Vector RQ = 5x
To solve for x, you must consider the relationships and constraints provided by the geometry and vector magnitudes.
For this problem, we must use the polygon law of vector addition. By summing all the vectors in a closed polygon, the resultant vector should be zero. This gives us the equation:
\[ PQ + QP + PT + TS + SR + RQ = 0 \]
Let's put all vectors in the appropriate direction and sum their magnitudes:
\[ 4x + 3x + 3x + 2x + x + 5x = 0 \]
Simplify the equation:
\[ 18x = 0 \]
Which suggests that the algebraic sum of magnitudes along each dimension should cancel out. Given that 18x cannot be zero for non-trivial cases, we assume:
\[ x = 0 \]
However, realizing that for a geometric interpretation 'x' cannot equate to zero, the vectors should form a closed shape confirming all vector sums per given direction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F491a39fc-a698-4751-b086-3ab1bda721ab%2Fc48818b8-f0cd-47a2-93be-0325fb74b8e6%2Ftja11e_processed.png&w=3840&q=75)
Transcribed Image Text:**Solve for X**
The following diagram shows a geometric shape with several vectors labeled. The vectors are defined with magnitudes that are multiples of a variable, x. The vectors are as follows:
- Vector PQ = 4x
- Vector QP = 3x
- Vector PT = 3x
- Vector TS = 2x
- Vector SR = x
- Vector RQ = 5x
To solve for x, you must consider the relationships and constraints provided by the geometry and vector magnitudes.
For this problem, we must use the polygon law of vector addition. By summing all the vectors in a closed polygon, the resultant vector should be zero. This gives us the equation:
\[ PQ + QP + PT + TS + SR + RQ = 0 \]
Let's put all vectors in the appropriate direction and sum their magnitudes:
\[ 4x + 3x + 3x + 2x + x + 5x = 0 \]
Simplify the equation:
\[ 18x = 0 \]
Which suggests that the algebraic sum of magnitudes along each dimension should cancel out. Given that 18x cannot be zero for non-trivial cases, we assume:
\[ x = 0 \]
However, realizing that for a geometric interpretation 'x' cannot equate to zero, the vectors should form a closed shape confirming all vector sums per given direction.
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