Problem 60 - Precalculus with Limits

Question

Vector Operations In Exercises \(57-62\) , find the component form of \(v\) and sketch the specified vector operations geometrically, where \(u=2 i-j\) and \(w=i+2 j\) $$\mathbf{v}=-\mathbf{u}+\mathbf{w}$$

Step-by-Step Solution

Answer

The component form of \(v\) is \([-1, 3]\) or in coordinates it is \(-i + 3j\).

Step 1: Understand Vectors

Vectors \(u\) and \(w\) are given as \(u = 2i - j\) and \(w = i + 2j\). These vectors can also be represented in component form as \(u = [2, -1]\) and \(w = [1, 2]\). In component form, the first element represents the coefficient of \(i\) (x-coordinate), and the second element represents the coefficient of \(j\) (y-coordinate).

Step 2: Compute -u

The negative of a vector is found by switching the signs of each of its components. Hence, \(-u\) would be \([-2, 1]\).

Step 3: Compute sum of vectors

The operation specified is \(v = -u + w\). Now that we have computed \(-u\), we can add it to \(w\). In component form, vector addition involves adding corresponding components together. So, \(v = -u + w = [-2, 1] + [1, 2] = [-2+1, 1+2] = [-1, 3]\).

Step 4: Convert to coordinates

Now that we have computed the vector \(v\) in component form as \([-1, 3]\), we can convert it into coordinates. Thus, \(v = -i + 3j\).

Key Concepts

Component Form

Component form is a way to express a vector using a pair of numbers that represent its influence in the horizontal and vertical directions, typically aligned with the x and y axes. For example, a vector \( u \) is expressed in terms of \( i \) and \( j \), where \( i \) is the unit vector along the x-axis and \( j \) is the unit vector along the y-axis. This is often represented in component form as \([x, y]\).

To convert the vector \( u = 2i - j \) to component form, identify the coefficients of \( i \) and \( j \):

• \( 2 \) for \( i \)
• \( -1 \) for \( j \)

Thus, \( u \) is expressed in component form as \([2, -1]\). Similarly, for \( w = i + 2j \), the component form is \([1, 2]\).

This format simplifies arithmetic operations and helps in visualizing vectors in a coordinate system.

Vector Addition

Vector addition is the process of summing two or more vectors, resulting in another vector. This operation maintains the properties of vector space, where summing is simply aligning the original vectors tail-to-head and finding the resultant.

To add vectors in component form, simply add the corresponding components.

For the vectors \( -u = [-2, 1] \) and \( w = [1, 2] \), vector addition works as follows:

• Add the x-components: \(-2 + 1 = -1\)
• Add the y-components: \(1 + 2 = 3\)

This results in the vector \( v = [-1, 3] \).

This method of adding components individually ensures accuracy and makes complex vector operations more manageable.

Geometric Representation

Geometric representation of vectors provides a visual understanding of vectors and vector operations. It plots vectors as arrows on a graph, showing direction and magnitude.

When geometrically representing the vector \( v = -u + w \), start by drawing vectors \( u \) and \( w \) based on their components:

• \( u = [2, -1] \) begins at any point and moves 2 units in the positive x-direction and 1 unit in the negative y-direction.
• \( w = [1, 2] \) begins at the same initial point as \( u \) and moves 1 unit in the positive x-direction and 2 units in the positive y-direction.

Next, draw the vector \(-u\) by reversing \( u \)'s direction, moving 2 units in the negative x-direction and 1 unit in the positive y-direction.

Finally, add \(-u\) and \(w\) by placing the tail of \(w\) at the head of \(-u\) and drawing the resultant \( v \) from the tail of \(-u\) to the head of \(w\). The vector \( v = [-1, 3] \) can thus be visualized geometrically, offering insight into both the magnitude and direction of the resulting vector.