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Chapter 7: Problem 60

Express the vector as a sum of unit vectors \(i\) and \(\mathbf{j}\). $$ \langle 0,2\rangle $$

Short Answer

Expert verified
The vector \( \langle 0, 2 \rangle \) is expressed as \( 0\mathbf{i} + 2\mathbf{j} \), simplifying to \( 2\mathbf{j} \).

Step by step solution

01

Identifying Vector Components

The given vector is \( \langle 0, 2 \rangle \). This means the first component of the vector is 0, and the second component is 2.
02

Applying Unit Vectors

Recall that the unit vector \( \mathbf{i} \) represents the horizontal direction and is written as \( \langle 1, 0 \rangle \). Similarly, the unit vector \( \mathbf{j} \) represents the vertical direction and is written as \( \langle 0, 1 \rangle \).
03

Rewriting the Vector

To express \( \langle 0, 2 \rangle \) as a sum of unit vectors, identify how it can be decomposed into multiples of \( \mathbf{i} \) and \( \mathbf{j} \). The vector can be written as \( 0 \times \mathbf{i} + 2 \times \mathbf{j} \).
04

Constructing the Final Expression

Since \( 0 \times \mathbf{i} \) is the zero vector, the simplified expression for the vector \( \langle 0, 2 \rangle \) remains \( 2\mathbf{j} \). Therefore, the vector can be fully described as \( 2\mathbf{j} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
When dealing with vectors, understanding their components is crucial. Each vector in a two-dimensional space can be broken down into two parts: the horizontal component and the vertical component. These components determine the direction and size of a vector. To represent a vector like \( \langle 0, 2 \rangle \), we refer to these parts. Here, the first component is 0, meaning it has no horizontal influence. The second component is 2, which signifies that the vector has a vertical influence along the y-axis.
  • Horizontal component: This is the x-value in \( \langle x, y \rangle \), which defines the vector鈥檚 movement on the x-axis.
  • Vertical component: This is the y-value in \( \langle x, y \rangle \), indicating the vector鈥檚 movement on the y-axis.
Each vector can thus be described by these components, showing where and how the vector moves in space.
Horizontal Direction
The horizontal direction in vector mathematics is represented by unit vector \( \mathbf{i} \). Each unit vector has a length of 1 and is used to suggest direction without affecting magnitude. In a general vector \( \langle x, y \rangle \), the horizontal movement is captured purely by the x-component.
  • Unit vector \( \mathbf{i} \) is expressed as \( \langle 1, 0 \rangle \), pointing exclusively along the x-axis.
  • Vectors multiplied by \( \mathbf{i} \) affect only the horizontal direction, as seen in \( 0 \times \mathbf{i} \) leading to the null vector in the x-direction.
  • For \( \langle 0, 2 \rangle \), the x-component and thus the horizontal movement is 0, exemplifying no movement horizontally.
Understanding this simplifies working with vectors in practical scenarios, letting students break down more complex vectors.
Vertical Direction
In vector spaces, the vertical direction is indicated using the unit vector \( \mathbf{j} \). Similar to the horizontal unit vector, \( \mathbf{j} \) also possesses a magnitude of 1, but correlates to movement along the y-axis. The vertical component of a vector is its y-coordinate, reflecting how the vector moves upwards or downwards.
  • Unit vector \( \mathbf{j} \) is represented as \( \langle 0, 1 \rangle \), serving solely to indicate vertical direction.
  • For the vector \( \langle 0, 2 \rangle \), multiplying 2 by \( \mathbf{j} \) translates to \( 2\mathbf{j} \), clearly demonstrating two units of movement vertically.
  • This decomposition using unit vectors allows the strategic and straightforward expression of vectors in componential form.
By understanding vertical direction in this structured way, dealing with vector expressions becomes intuitive.

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Most popular questions from this chapter

A force of 1000 pounds is acting on an object at an angle of \(45^{\circ}\) from the horizontal. Another force of 500 pounds is acting at an angle of \(-40^{\circ}\) from the horizontal. What is the magnitude of the resultant force?

Let \(\mathbf{u}=\langle-2,3\rangle, \mathbf{v}=\langle 4,1\rangle\), and \(\mathbf{w}=\langle-3,-5\rangle\). Demonstrate that \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\).

In Exercises 1-16, find each of the following dot products. $$ \langle 4,-2\rangle \cdot\langle 3,5\rangle $$

Determine whether each pair of vectors is orthogonal. $$ \langle 0.8,4\rangle \text { and }\langle 3,-6\rangle $$

Use a calculator to perform the vector operation given \(\mathbf{u}=\langle 8,-5\rangle\) and \(\mathbf{v}=\langle-7,11\rangle\). $$ \mathbf{u}+3 \mathbf{v} $$
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