Problem 58 - 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 \(w=i+2 j\) $$\mathbf{v}=\frac{3}{4} \mathbf{w}$$

Step-by-Step Solution

Answer

The vector \(v\) is \(\frac{3}{4}i + \frac{3}{2}j\). Geometrically, it can sketched as a vector beginning at the origin and ending at the point (\frac{3}{4}, \frac{3}{2}).

Step 1: Find the scaled vector

The first step to solve the exercise is to find the vector v. As told in the problem, the vector \(v\) is \(\frac{3}{4}\) times the vector \(w\). Therefore, we can find the vector \(v\) by multiplying each component of the vector \(w\) with \(\frac{3}{4}\). Let's do that.

Step 2: Calculate vector v

The vector \(w\) is given as \(i + 2j\). Therefore, to calculate the vector \(v\), we can multiply each component of \(w\) with \(\frac{3}{4}\). This would give us: \(v = \frac{3}{4}i + \frac{3}{4} * 2j = \frac{3}{4}i + \frac{3}{2}j\).

Step 3: Sketch the vector operations geometrically

Now that we have calculated vector \(v\), the last step is to sketch the vectors \(v\) and \(w\) geometrically. In this sketch, both vectors should start from the origin. Draw \(w\) as a vector starting at the origin and ending at point (1,2). Next, draw \(v\) also starting at the origin which ends at point (\frac{3}{4}, \frac{3}{2}).

Key Concepts

Component Form of a Vector

Understanding the component form of a vector is crucial for performing operations such as addition, subtraction, and scaling in vector analysis. A vector is an entity that has both magnitude and direction. In a two-dimensional space, a vector is defined by its horizontal (x) and vertical (y) components. The component form of a vector is expressed as \(\mathbf{v} = x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the horizontal and vertical directions respectively.

The task in the exercise was to find the component form of the vector \(\mathbf{v}\) given that it is a scaled version of vector \(\mathbf{w}\). To find the components of \(\mathbf{v}\), we can multiply the scalar, \(\frac{3}{4}\), with each component of \(\mathbf{w}\), resulting in \(\mathbf{v} = \frac{3}{4}\mathbf{i} + \frac{3}{2}\mathbf{j}\). This means the vector \(\mathbf{v}\) has a horizontal component of \(\frac{3}{4}\) and a vertical component of \(\frac{3}{2}\).

Vector Scaling

The process of vector scaling involves multiplying a vector by a scalar (a real number), which changes the magnitude of the vector without altering its direction. This operation is also known as scalar multiplication. When you scale a vector, each of its components is multiplied by the same scalar value.

In the given exercise, vector scaling was demonstrated by multiplying vector \(\mathbf{w}\) by the scalar \(\frac{3}{4}\). Scaling affects only the magnitude, making vector \(\mathbf{v}\) shorter or longer depending on whether the scalar is less than or greater than one. With \(\mathbf{w} = \mathbf{i} + 2\mathbf{j}\), after scaling we got \(\mathbf{v} = \frac{3}{4}\mathbf{i} + \frac{3}{2}\mathbf{j}\), which is simply the vector \(\mathbf{w}\) shortened to three-fourths of its original length. This technique is widely used to resize vectors in physics, engineering, and computer graphics.

Geometric Representation of Vectors

Vectors can be represented geometrically in a coordinate system where they are drawn as directed line segments. The direction indicates the direction of the vector, and the length (or magnitude) of the line segment represents the magnitude of the vector.

In our origin exercise, we needed to sketch vector \(\mathbf{v}\) geometrically alongside vector \(\mathbf{w}\). This involves plotting them on a graph with their tails at the origin (0,0). Vector \(\mathbf{w}\) ends at point (1,2), and because vector \(\mathbf{v}\) is a scaled version of \(\mathbf{w}\), it ends at point \(\left(\frac{3}{4}, \frac{3}{2}\right)\), indicating that it's shorter than \(\mathbf{w}\) yet pointing in the same direction. This visual representation helps students to better grasp the effects of operations like scaling on the vector's magnitude and direction, and it provides an intuitive understanding of vector arithmetic.