Problem 62 - 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 \(w=i+2 j\) $$\mathbf{v}=\mathbf{u}-2 \mathbf{w}$$

Step-by-Step Solution

Answer

The component form of the vector \(v = u - 2w\) is \(v = -5j\), which indicates that the vector is directed along the y-axis towards the negative direction. Geometrically plotted, vector \(v\) starts at the origin and points downwards on the y-axis till the point (0,-5).

Step 1: Substitute the values of vectors u and w

First let's substitute the given vectors \(u=2i-j\) and \(w=i+2j\) into the equation \(v=u-2w\). So it becomes: \(v = (2i-j) - 2*(i+2j)\). Let's simplify this expression to find out the component form of vector v.

Step 2: Simplify the expression to find the vector's component form

After substituting the values of vectors u and w into our equation, we have to simplify the expression. This expression simplifies to: \(v = (2i-j) - 2i - 4j = 2i - 2i - j - 4j = -5j\). This means that vector v lies entirely along the y-axis, and that it is directed downwards because its coefficient, -5, is negative.

Step 3: Plotting the vectors

To plot the vectors geometrically, we can place them into a 2D coordinate system because they have two components i and j which can be identified with x and y axes correspondingly. Vector \(u = 2i-j\) is located at (2,-1), vector \(w = i+2j\) is located at (1,2), and the resulting vector \(v = -5j\) lies along the y-axis and points downwards from the origin (0,-5).

Key Concepts

Component Form of a Vector

Understanding the component form of a vector is essential for performing vector operations. Vectors are entities that have both magnitude and direction. In two dimensions, they can be expressed in terms of two perpendicular components, typically along the x-axis (i) and y-axis (j).
For instance, the vector u given by u = 2i - j can be described as having an x-component of 2 and a y-component of -1. In this form, it’s straightforward to manipulate vectors algebraically, just like we do with real numbers.
When we add or subtract vectors, we simply combine or subtract their corresponding components. This leads us to another key operation - vector subtraction.

Vector Subtraction

Vector subtraction is a fundamental operation that involves taking the components of one vector away from another. When we see an equation like v = u - 2w, it signifies that we should multiply the components of vector w by 2 and subtract from the vector u.

Subtraction Process

The steps to perform this operation are simple:

• Identify the components of each vector.
• Double the components of vector w since we have 2w.
• Subtract the components of 2w from u.

For our case, the subtraction transforms into (2i - j) - 2(i + 2j), which simplifies to -5j. Notice that the resultant vector is solely in the direction of j (y-component), indicating a vertical vector in the coordinate system.
The manner in which we subtract these vectors is similar to handling numerical subtraction, except that we deal with each component separately.

Plotting Vectors in a Coordinate System

Plotting vectors in a coordinate system allows us to visualize the magnitude and direction of vectors. A coordinate system is a plane with a horizontal x-axis and a vertical y-axis. Each vector is represented as an arrow starting from the origin (0,0) and ending at a point defined by its components.

Vector Coordinates

In our example, the vector u has its head at the point (2, -1) and vector w at the point (1, 2), which means you move from the origin right for positive x values, left for negative x values, up for positive y values, and down for negative y values to draw the vector.

• For u, move 2 units right and 1 unit down from the origin.
• For w, move 1 unit right and 2 units up from the origin.

Vector v, which is the subtraction result, will point downwards along the y-axis and be located at (0, -5) since it has no x-component and has a y-component of -5.
Plotting vectors gives us an immediate visual understanding of the relation between vectors, such as the direction in which they point and how they combine to form resultant vectors.