Math Problem Statement

what is teh angle between the vector i + 2j + 3k and the x axis

Solution

To find the angle between the vector v=i+2j+3k\mathbf{v} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} and the xx-axis, we use the dot product formula.

Given:

  • Vector v=i+2j+3k\mathbf{v} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
  • Unit vector along the xx-axis is i\mathbf{i} (which can be written as i+0j+0k\mathbf{i} + 0\mathbf{j} + 0\mathbf{k})

The dot product vi\mathbf{v} \cdot \mathbf{i} is given by: vi=(1)(1)+(2)(0)+(3)(0)=1\mathbf{v} \cdot \mathbf{i} = (1)(1) + (2)(0) + (3)(0) = 1

The magnitude of v\mathbf{v} is: v=12+22+32=1+4+9=14|\mathbf{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}

The magnitude of i\mathbf{i} is: i=12+02+02=1|\mathbf{i}| = \sqrt{1^2 + 0^2 + 0^2} = 1

Now, the angle θ\theta between v\mathbf{v} and the xx-axis is found using the dot product formula: cosθ=vivi=114×1=114\cos \theta = \frac{\mathbf{v} \cdot \mathbf{i}}{|\mathbf{v}| |\mathbf{i}|} = \frac{1}{\sqrt{14} \times 1} = \frac{1}{\sqrt{14}}

So, the angle θ\theta is: θ=cos1(114)\theta = \cos^{-1}\left(\frac{1}{\sqrt{14}}\right)

This is the exact value of the angle. To express it as a decimal: θcos1(0.2673)74.5\theta \approx \cos^{-1}\left(0.2673\right) \approx 74.5^\circ

Summary:

The angle between the vector i+2j+3k\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} and the xx-axis is approximately 74.5 degrees.

Would you like more details or have any questions? Here are some related questions to explore:

  1. What is the angle between the vector and the yy-axis?
  2. How would you find the angle between two arbitrary vectors?
  3. Can the dot product be negative, and what does that signify?
  4. What are some properties of the dot product of two vectors?
  5. How would you calculate the projection of one vector onto another?

Tip: When calculating angles between vectors, always ensure both vectors are expressed in the same coordinate system!

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Math Problem Analysis

Mathematical Concepts

Vector Mathematics
Dot Product
Trigonometry

Formulas

Dot Product Formula: v · u = |v| |u| cos(θ)
Magnitude of a Vector: |v| = √(x^2 + y^2 + z^2)
Cosine Inverse: θ = cos^(-1)(x)

Theorems

Dot Product Theorem

Suitable Grade Level

Grades 10-12