Math Problem Statement
Solution
The text in the image contains a mathematical problem written in Arabic. Let me transcribe and analyze it:
Translation of the problem:
"If a vector forms angles with measures with the positive axes , , and respectively, and if the magnitude of is , find the vector ."
Steps to Solve:
-
Understand the Angles and Components:
- The given angles are:
- with the -axis,
- with the -axis,
- with the -axis.
- These angles represent the direction cosines of the vector. The direction cosines are computed as: Substituting the values:
- The given angles are:
-
Components of the Vector: The components of the vector are given by: Substituting and the cosine values:
-
Vector Form: The vector can be written in component form:
-
Simplify the Components:
- ,
- .
Therefore, the vector becomes:
Final Answer:
Let me know if you need further explanation or steps to verify the calculations!
5 Related Questions:
- What are direction cosines, and how are they derived?
- How do we calculate the magnitude of a vector from its components?
- What is the significance of a vector's angle with an axis?
- How can this problem be extended to find unit vectors?
- Can we generalize this problem for arbitrary angles?
Tip:
Always double-check trigonometric values (e.g., , ) to ensure accuracy in vector calculations!
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Math Problem Analysis
Mathematical Concepts
Vectors
Direction Cosines
Vector Magnitude
Formulas
cos(θx), cos(θy), cos(θz) are the direction cosines
rx = ||r|| * cos(θx), ry = ||r|| * cos(θy), rz = ||r|| * cos(θz)
Theorems
Direction Cosines of a Vector
Suitable Grade Level
Grades 11-12
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