Find A Vector Parallel To And With Magnitude .

Finding a Vector Parallel to Another with a Specified Magnitude

Finding a vector parallel to a given vector and possessing a specific magnitude is a fundamental concept in linear algebra and vector calculus with applications spanning diverse fields like physics, engineering, and computer graphics. This process involves understanding vector properties, specifically scalar multiplication and normalization. This article will comprehensively guide you through the process, clarifying the underlying principles and providing practical examples.

Understanding Vector Properties

Before diving into the solution, let's revisit some essential vector properties:

1. Parallel Vectors:

Two vectors are parallel if they are scalar multiples of each other. In other words, if vector v is parallel to vector u, then there exists a scalar 'k' such that:

v = ku

where k is any real number. If k > 0, the vectors point in the same direction; if k < 0, they point in opposite directions.

2. Magnitude (or Length) of a Vector:

The magnitude of a vector, denoted by ||v||, represents its length. For a vector v = (v_x, v_y, v_z) in three-dimensional space, the magnitude is calculated using the Pythagorean theorem in three dimensions:

||v|| = √(v_x² + v_y² + v_z²)

For a two-dimensional vector v = (v_x, v_y), the magnitude is:

||v|| = √(v_x² + v_y²)

3. Unit Vectors:

A unit vector is a vector with a magnitude of 1. It indicates a direction in space. Any non-zero vector can be normalized (converted into a unit vector) by dividing it by its magnitude:

û = v / ||v||

The Procedure: Finding a Parallel Vector with a Given Magnitude

The process of finding a vector parallel to a given vector and with a specified magnitude involves two key steps:

1. Normalization: Create a unit vector in the direction of the given vector.

2. Scalar Multiplication: Scale the unit vector to achieve the desired magnitude.

Let's break it down step-by-step:

Step 1: Given Vector and Desired Magnitude

Assume we have a given vector u = (u_x, u_y, u_z) and we want to find a parallel vector v with magnitude 'm'.

Step 2: Calculate the Magnitude of the Given Vector

Compute the magnitude of the given vector u using the formula mentioned earlier:

||u|| = √(u_x² + u_y² + u_z²)

Step 3: Normalize the Given Vector

To obtain a unit vector û in the same direction as u, divide each component of u by its magnitude:

û = (u_x/||u||, u_y/||u||, u_z/||u||)

Note: If the magnitude of u is zero (u is the zero vector), you cannot normalize it, as division by zero is undefined. Any vector parallel to the zero vector will also be the zero vector.

Step 4: Scale the Unit Vector to the Desired Magnitude

Now, multiply the unit vector û by the desired magnitude 'm' to obtain the parallel vector v with the specified magnitude:

v = mû = m(u_x/||u||, u_y/||u||, u_z/||u||) = (mu_x/||u||, mu_y/||u||, mu_z/||u||)

Step 5: Verification

To verify the result, calculate the magnitude of the vector v:

||v|| = √((mu_x/||u||)² + (mu_y/||u||)² + (mu_z/||u||)²) = √(m²(u_x²/||u||² + u_y²/||u||² + u_z²/||u||²)) = m√((u_x² + u_y² + u_z²)/||u||²) = m√(1) = m

This confirms that the magnitude of v is indeed 'm'. Furthermore, since v is a scalar multiple of u, it's parallel to u.

Examples

Let's illustrate this process with some examples.

Example 1: Two-Dimensional Vector

Find a vector v parallel to u = (3, 4) and with magnitude 10.

1. Magnitude of u: ||u|| = √(3² + 4²) = 5

2. Unit vector û: û = (3/5, 4/5)

3. Vector v: v = 10û = 10(3/5, 4/5) = (6, 8)

4. Verification: ||v|| = √(6² + 8²) = 10

Example 2: Three-Dimensional Vector

Find a vector v parallel to u = (1, 2, 2) and with magnitude 3.

1. Magnitude of u: ||u|| = √(1² + 2² + 2²) = 3

2. Unit vector û: û = (1/3, 2/3, 2/3)

3. Vector v: v = 3û = 3(1/3, 2/3, 2/3) = (1, 2, 2)

4. Verification: ||v|| = √(1² + 2² + 2²) = 3

In this case, the desired magnitude (3) matches the original vector's magnitude, resulting in the same vector. This is because the original vector already had the desired magnitude.

Example 3: Vector in the Opposite Direction

Find a vector v parallel to u = (1, -1) and with magnitude 5, but pointing in the opposite direction.

1. Magnitude of u: ||u|| = √(1² + (-1)²) = √2

2. Unit vector û: û = (1/√2, -1/√2)

3. Vector v (opposite direction): v = -5û = (-5/√2, 5/√2)

4. Verification: ||v|| = √((-5/√2)² + (5/√2)²) = 5

Applications

The ability to find a vector parallel to another with a given magnitude has numerous applications:

• Physics: Representing forces, velocities, and accelerations in a specific direction with a given magnitude.

• Computer Graphics: Scaling and manipulating vectors in 3D space for object transformations, lighting calculations, and camera positioning.

• Engineering: Analyzing forces and stresses on structures, designing pathways for robots, and managing fluid flow calculations.

Conclusion

Finding a vector parallel to a given vector with a specified magnitude is a crucial skill in various scientific and engineering disciplines. By following the steps of normalization and scalar multiplication outlined in this article, you can confidently tackle this problem and apply it in your respective fields. Remember to always verify your results to ensure the parallel vector indeed possesses the desired magnitude and direction. Understanding the underlying principles of vector algebra and the properties of unit vectors are vital for mastering this process and its many applications.