Math Problem Statement

Find the unit vector that has the same direction as v = 2i + 2j - k.

Solution

To find the unit vector in the same direction as v = 2i + 2j - k, we need to normalize this vector. The unit vector v^\hat{v} is given by the formula:

v^=vv\hat{v} = \frac{\vec{v}}{|\vec{v}|}

Where v|\vec{v}| is the magnitude of vector v\vec{v}, and it's calculated as:

v=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

For v=2i+2jk\vec{v} = 2i + 2j - k, we have:

  • vx=2v_x = 2
  • vy=2v_y = 2
  • vz=1v_z = -1

Now, calculate the magnitude:

v=22+22+(1)2=4+4+1=9=3|\vec{v}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3

Now, divide each component of the vector by the magnitude to get the unit vector:

v^=13(2i+2jk)=23i+23j13k\hat{v} = \frac{1}{3}(2i + 2j - k) = \frac{2}{3}i + \frac{2}{3}j - \frac{1}{3}k

Thus, the unit vector in the same direction as v is:

v^=23i+23j13k\hat{v} = \frac{2}{3}i + \frac{2}{3}j - \frac{1}{3}k

Would you like more details, or have any questions on this?


Related Questions:

  1. What is the formula for finding the unit vector in general?
  2. How do you calculate the magnitude of a vector in 3D space?
  3. Can the direction of a vector be changed by normalizing it?
  4. What are the applications of unit vectors in physics and engineering?
  5. How do you compute the dot product between two vectors?

Tip: The unit vector is often used to indicate direction without affecting magnitude, making it especially useful in physics problems involving direction or force.

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Math Problem Analysis

Mathematical Concepts

Vector Algebra
Unit Vectors
3D Vectors

Formulas

Unit vector formula: \(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\)
Magnitude formula for 3D vector: \(|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\)

Theorems

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Suitable Grade Level

Grade 10-12