Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\).$$\mathbf{v}=\frac{3}{4} \mathbf{w}$$

Step-by-Step Solution

Answer

The component form of \(v\) is \((0.75, 1.5)\).

Step 1: Find the Component Form of \(w\)

The given vector \(\mathbf{w}\) has its components \(\mathbf{i}\) and \(\mathbf{j}\) given. So we will express \(\mathbf{w}\) in component form. In general, \(Ai + Bj\) refers to \([A, B]\) in component form. Therefore, \(\mathbf{w} = \mathbf{i}+2\mathbf{j}\) can be written as (1,2).

Step 2: Calculate \(v\)

Now, apply the given formula \(\mathbf{v}=\frac{3}{4} \mathbf{w}\). Multiply the components of \(w\) by \(\frac{3}{4}\) which gives \((\frac{3}{4}*1, \frac{3}{4}*2)\) reducing to \((0.75, 1.5)\). This is the component form of \(v\).

Step 3: Sketch Vector v

On a graph, we start at the origin (0,0) and draw a line to the point \((0.75, 1.5)\) according to the component of vector \(v\). This line represents our vector \(v\).

Key Concepts

Vector Operations

Understanding vector operations is critical in fields like physics, engineering, and computer science. Vectors are mathematical entities that have both magnitude and direction, and they represent quantities such as displacement, velocity, or force. To perform operations on vectors such as addition, subtraction, and scalar multiplication, it is often necessary to express them in a form that isolates each of their individual components.

In the context of our exercise, vector operations like scaling, which involves multiplying a vector by a scalar (a real number), require handling each component of the vector separately. Scalar multiplication, such as the operation \( \mathbf{v}=\frac{3}{4} \mathbf{w} \), affects both the magnitude and direction of a vector but preserves the angle it makes with the axes. In other words, when we scale \( \mathbf{w} \) to get \( \mathbf{v} \) by \( \frac{3}{4} \) we're shrinking the length of \( \mathbf{w} \) to three-fourths of its original length, but it's still pointing in the same direction.

Component Form Calculation

The component form of a vector is a convenient way to describe its properties using an ordered pair (for two dimensions) or triplet (for three dimensions), representing how much the vector stretches along each axis. To calculate the component form of a vector, you need to identify its horizontal and vertical components (and the third, if dealing with three dimensions).

In our example, \( \mathbf{w} \) is originally in the form \( \mathbf{i}+2\mathbf{j} \), which tells us it has a component of 1 in the horizontal (x) direction and a component of 2 in the vertical (y) direction, translating into the component form of (1, 2). When we are given the operation \( \mathbf{v}=\frac{3}{4} \mathbf{w} \), we multiply each component of \( \mathbf{w} \) by \( \frac{3}{4} \) to acquire the new components, resulting in \( (0.75, 1.5) \) for \( \mathbf{v} \). This process is fundamental when performing vector operations, as it allows for a clear view of how vectors interact with each other along each axis.

Vector Sketching

Sketching vectors is a powerful visual tool that aids in understanding their behavior in a given space. To sketch a vector, one often uses a coordinate system where the vector is represented as an arrow pointing from the origin to a specific point, which corresponds to the vector's components. This is particularly useful when dealing with operations such as vector addition, where the resultant vector can be visualized as the diagonal of a parallelogram constructed by the original vectors.

In the context of the exercise, vector \( \mathbf{v} \) is sketched from the origin to the point \( (0.75, 1.5) \), which corresponds to its component form. The point (0.75, 1.5) symbolizes how far along the x-axis and y-axis the vector reaches. Starting at the origin, you move 0.75 units horizontally and 1.5 units vertically to reach the end of the vector. This visual approach is not only helpful in understanding individual vector components but also provides insight into the result of vector operations, like scaling in our example.