Math Problem Statement

If a vector r forms angles with measures (0°, 90°, 60°) with the positive axes x, y, and z respectively, and if the magnitude of r is ||r|| = √40, find the vector r.

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 r\vec{r} forms angles with measures (0°,90°,60°)(0°, 90°, 60°) with the positive axes xx, yy, and zz respectively, and if the magnitude of r\vec{r} is r=40||\vec{r}|| = \sqrt{40}, find the vector r\vec{r}."


Steps to Solve:

  1. Understand the Angles and Components:

    • The given angles are:
      • 00^\circ with the xx-axis,
      • 9090^\circ with the yy-axis,
      • 6060^\circ with the zz-axis.
    • These angles represent the direction cosines of the vector. The direction cosines are computed as: cos(θx)=cos(0),cos(θy)=cos(90),cos(θz)=cos(60).\cos(\theta_x) = \cos(0^\circ), \quad \cos(\theta_y) = \cos(90^\circ), \quad \cos(\theta_z) = \cos(60^\circ). Substituting the values: cos(0)=1,cos(90)=0,cos(60)=12.\cos(0^\circ) = 1, \quad \cos(90^\circ) = 0, \quad \cos(60^\circ) = \frac{1}{2}.
  2. Components of the Vector: The components of the vector r\vec{r} are given by: rx=rcos(θx),ry=rcos(θy),rz=rcos(θz).r_x = ||\vec{r}|| \cdot \cos(\theta_x), \quad r_y = ||\vec{r}|| \cdot \cos(\theta_y), \quad r_z = ||\vec{r}|| \cdot \cos(\theta_z). Substituting r=40||\vec{r}|| = \sqrt{40} and the cosine values: rx=401=40,ry=400=0,rz=4012=402.r_x = \sqrt{40} \cdot 1 = \sqrt{40}, \quad r_y = \sqrt{40} \cdot 0 = 0, \quad r_z = \sqrt{40} \cdot \frac{1}{2} = \frac{\sqrt{40}}{2}.

  3. Vector Form: The vector r\vec{r} can be written in component form: r=(40,0,402).\vec{r} = \left(\sqrt{40}, 0, \frac{\sqrt{40}}{2}\right).

  4. Simplify the Components:

    • 40=410=210\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10},
    • 402=2102=10\frac{\sqrt{40}}{2} = \frac{2\sqrt{10}}{2} = \sqrt{10}.

    Therefore, the vector becomes: r=(210,0,10).\vec{r} = \left(2\sqrt{10}, 0, \sqrt{10}\right).


Final Answer:

r=(210,0,10)\vec{r} = \left(2\sqrt{10}, 0, \sqrt{10}\right)


Let me know if you need further explanation or steps to verify the calculations!


5 Related Questions:

  1. What are direction cosines, and how are they derived?
  2. How do we calculate the magnitude of a vector from its components?
  3. What is the significance of a vector's angle with an axis?
  4. How can this problem be extended to find unit vectors?
  5. Can we generalize this problem for arbitrary angles?

Tip:

Always double-check trigonometric values (e.g., cos\cos, sin\sin) 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