Problem 28 - Algebra and Trigonometry

Question

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle- 2,10\rangle $$

Step-by-Step Solution

Answer

\(-2\mathbf{i} + 10\mathbf{j}\).

Step 1: Identify Components

Identify the components of the given vector \(\mathbf{u} = \langle -2, 10 \rangle \). The vector \(\mathbf{u}\) has an x-component of -2 and a y-component of 10.

Step 2: Express in Terms of Vectors

Express the given vector \(\mathbf{u}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). The unit vector \(\mathbf{i}\) represents the x-direction and the unit vector \(\mathbf{j}\) represents the y-direction.

Step 3: Write the Final Expression

Write each component of the vector as a product with its corresponding unit vector: \(-2\mathbf{i} + 10\mathbf{j} \). This expresses the vector \(\mathbf{u}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

Key Concepts

Vector components

Vectors are mathematical entities that have both magnitude and direction. When you look at a vector like \( \mathbf{u} = \langle -2, 10 \rangle \), it's described in terms of its components. Components are the building blocks of any vector, and they tell us how far the vector travels along each axis of a coordinate system.- The **x-component** refers to how much the vector moves left or right along the horizontal axis. In \( \mathbf{u} = \langle -2, 10 \rangle \), the x-component is \(-2\).- The **y-component** represents movement up or down along the vertical axis. For vector \( \mathbf{u} \), the y-component is \(10\).Each component reflects a part of the vector's overall movement, and you can visualize them as projections of the vector on the axes. By understanding components, you can easily reconstitute the whole vector, which is crucial in physics and engineering where direction matters along each axis.

Unit vectors

Understanding unit vectors is essential in vector mathematics, as they serve as reference vectors that point in particular directions. Unit vectors have a magnitude of exactly 1 and provide direction-specific components for constructing vectors.- **Unit vector \( \mathbf{i} \)** denotes the unit vector along the x-axis. Its function is to quantify the direction and extent of movement along this axis without affecting the magnitude of the vector.- **Unit vector \( \mathbf{j} \)** is similar but applies along the y-axis, representing vertical components.In our example, vector \( \mathbf{u} = \langle -2, 10 \rangle \) is broken down into components in terms of these unit vectors: - The x-component, \(-2\), pairs with \( \mathbf{i} \) to give \(-2\mathbf{i}\).- The y-component, \(10\), pairs with \( \mathbf{j} \) to become \(10\mathbf{j}\).By using these unit vectors, it simplifies the process of working with vectors as you can always break them down into their basic directional components.

Vector notation

Vectors can be represented in different notations, but the one involving unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) is very common. This notation is not only precise but also helps convey both the components and their directional influence in a clear manner.For vector \( \mathbf{u} = \langle -2, 10 \rangle \), we express it as \(-2\mathbf{i} + 10\mathbf{j}\). This gives a complete picture of the vector’s action:- **\(-2\mathbf{i}\)** indicates a movement 2 units to the left along the x-axis.- **\(+10\mathbf{j}\)** means the vector moves 10 units up along the y-axis.This notation is advantageous as it maintains consistency and clarity across various calculations, including vector addition, subtraction, and more complex operations like dot and cross products. By adopting this form, you'll find it easier to visualize and manipulate vectors within mathematical situations or real-world scenarios.