Finding A Vector To Achieve A Unit Vector Resultant

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In this article, we delve into the world of vector operations, specifically focusing on vector addition, scalar multiplication, and the concept of a unit vector. Our main objective is to find a vector w that, when added to the resultant vector of u + 3v, yields the unit vector <1, 0>. This problem combines several fundamental vector operations, making it an excellent exercise for understanding vector manipulation in a two-dimensional space.

Understanding Vectors and Vector Operations

Before we dive into the solution, let's briefly review some key concepts.

A vector is a quantity that has both magnitude (length) and direction. In a two-dimensional plane, a vector can be represented as an ordered pair of numbers, such as <x, y>, where x and y are the components of the vector along the horizontal and vertical axes, respectively. Vector operations are fundamental mathematical operations that manipulate vectors. The main vector operations are addition, subtraction, and scalar multiplication. Understanding vector operations is crucial for solving problems involving forces, velocities, and other physical quantities.

  • Vector Addition: To add two vectors, you add their corresponding components. For example, if u = <u₁, u₂> and v = <v₁, v₂>, then u + v = <u₁ + v₁, u₂ + v₂>.
  • Scalar Multiplication: To multiply a vector by a scalar (a real number), you multiply each component of the vector by that scalar. For example, if k is a scalar and u = <u₁, u₂>, then ku = <ku₁, ku₂>.
  • Unit Vector: A unit vector is a vector with a magnitude (length) of 1. The unit vector in the direction of the positive x-axis is typically denoted as <1, 0>, and the unit vector in the direction of the positive y-axis is denoted as <0, 1>. Unit vectors are essential for normalizing vectors and representing directions in space. The unit vector <1, 0> is particularly important as it serves as a reference for other vectors in the plane.

Problem Statement

Given two vectors, u = <-7, 6> and v = <-4, 17>, we need to find a vector w such that adding w to the resultant vector of u + 3v gives us the unit vector <1, 0>. This problem involves a combination of scalar multiplication, vector addition, and the concept of a unit vector. It's a great exercise in understanding vector operations and how they interact.

Mathematically, this can be expressed as:

u + 3v + w = <1, 0>

Our goal is to find the vector w that satisfies this equation. This involves performing scalar multiplication on vector v, adding the result to vector u, and then finding the vector w that, when added to this sum, results in the unit vector <1, 0>.

Step-by-Step Solution

To find the vector w, we will follow these steps:

  1. Calculate 3v: Multiply the vector v by the scalar 3.
  2. Calculate u + 3v: Add the result from step 1 to the vector u.
  3. Find w: Determine the vector w that, when added to the result from step 2, gives the unit vector <1, 0>.

Let's go through each step in detail:

1. Calculate 3v

We are given v = <-4, 17>. To find 3v, we multiply each component of v by 3:

3v = 3 * <-4, 17> = <3 * -4, 3 * 17> = <-12, 51>

So, 3v = <-12, 51>. This step involves the scalar multiplication of a vector, which is a fundamental vector operation.

2. Calculate u + 3v

Now we add the vector u = <-7, 6> to the result we obtained in step 1, which is 3v = <-12, 51>:

u + 3v = <-7, 6> + <-12, 51> = <-7 + (-12), 6 + 51> = <-19, 57>

So, u + 3v = <-19, 57>. This step involves vector addition, another key vector operation.

3. Find w

We want to find a vector w such that u + 3v + w = <1, 0>. We know that u + 3v = <-19, 57>. Let w = <x, y>. Then the equation becomes:

<-19, 57> + <x, y> = <1, 0>

Adding the vectors on the left side, we get:

<-19 + x, 57 + y> = <1, 0>

For these two vectors to be equal, their corresponding components must be equal. Therefore, we have two equations:

  1. -19 + x = 1
  2. 57 + y = 0

Solving for x in the first equation:

x = 1 + 19 = 20

Solving for y in the second equation:

y = -57

Thus, the vector w is <20, -57>.

Final Answer and Conclusion

The vector w that can be added to u + 3v to get the unit vector <1, 0> is w = <20, -57>. This problem demonstrates the importance of understanding vector operations, including scalar multiplication and vector addition, in solving vector-related problems. By breaking down the problem into smaller steps and applying the definitions of these operations, we were able to find the required vector w.

In conclusion, mastering vector operations is crucial for various applications in mathematics, physics, and engineering. This example provides a solid foundation for tackling more complex problems involving vectors and their manipulations.