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Chapter 8: Problem 69

Write each vector in the form a\mathbf{i} \(+b \mathbf{j}\) $$\langle 2,0\rangle$$

Short Answer

Expert verified
2\(\mathbf{i}\)

Step by step solution

01

Identify Components

Observe the given vector \(\langle 2, 0 \rangle\). It has two components: the first component is 2 and the second component is 0.
02

Assign Components to \(\mathbf{i}\) and \(\mathbf{j}\)

Assign the first component (2) to \(\mathbf{i}\) and the second component (0) to \(\mathbf{j}\).
03

Write in Vector Form

Write the vector in the form \(\mathbf{a} \mathbf{i} + \mathbf{b} \mathbf{j}\), where \(\mathbf{a} = \) 2 and \(\mathbf{b} = \) 0. So, the vector is \(\mathbf{2i} + 0 \mathbf{j}\) or simply \(\mathbf{2i}\).

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Key Concepts

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

vector components
Vectors are quantities that have both magnitude and direction. They can be broken down into smaller parts called components.

For a given vector \[\langle 2,0 \rangle\], the two components are 2 and 0.
These components represent how much the vector moves in the horizontal and vertical directions, respectively.
  • The first number (2) is the horizontal component.
  • The second number (0) is the vertical component.

Breaking a vector down into its components helps us understand its direction and length more clearly.
basis vectors
Basis vectors are standard vectors used to represent any vector in a given space.

In 2-dimensional space, the most common basis vectors are \mathbf{i}\ and \mathbf{j}\.
  • \mathbf{i}\ represents a unit vector in the horizontal direction (x-axis).
  • \mathbf{j}\ represents a unit vector in the vertical direction (y-axis).

By using basis vectors, we can represent any vector as a combination of \mathbf{i}\ and \mathbf{j}\. For example, the vector \[\langle 2, 0 \rangle\] can be written as \[2 \mathbf{i}\ + 0 \mathbf{j}\].
vector representation
Vector representation is the way we express vectors in terms of their components and basis vectors.

To represent the vector \[\langle 2, 0 \rangle\], we first identify its components (2 and 0).
Then, we assign these components to the corresponding basis vectors: 2 to \mathbf{i}\ and 0 to \mathbf{j}\.
  • The final representation becomes 2 \mathbf{i}\ + 0 \mathbf{j}\.

This method helps us visualize and work with vectors more effectively in mathematical and physical problems.

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Most popular questions from this chapter

For each pair of polar coordinates, ( \(a\) ) plot the point, ( \(b\) ) give two other pairs of polar coordinates for the point, and ( \(c\) ) give the rectangular coordinates for the point. $$\left(4, \frac{3 \pi}{2}\right)$$

Use a graphing calculator window of \([-1250,1250]\) by \([-1250,1250],\) in degree mode, to graph more of \(r=2 \theta\) (a spiral of Archimedes) than what is shown in Figure \(71 .\) Use \(-1250^{\circ} \leq \theta \leq 1250^{\circ} .\)

Give a complete graph of each polar equation. Also identify the type of polar graph. $$r=4 \cos 2 \theta$$

For each equation, find an equivalent equation in rectangular coordinates, and graph. $$r=\frac{3}{1-\sin \theta}$$

Give a complete graph of each polar equation. Also identify the type of polar graph. $$r=3+\cos \theta$$
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