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Chapter 5: Problem 29

If \(\mathbf{u}=3 \mathbf{i}-\mathbf{j}\) and \(\mathbf{v}=2 \mathbf{i}+4 \mathbf{j},\) find the horizontal and the vertical components of the indicated vector. $$ 2 \mathbf{u}-\mathbf{v} $$

Short Answer

Expert verified
The horizontal component is 4, and the vertical component is -6.

Step by step solution

01

Understanding Vectors

Vectors are expressions that represent quantities having both magnitude and direction. The vector \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \) consists of a horizontal component \( 3 \mathbf{i} \) and a vertical component \( - \mathbf{j} \), while the vector \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \) consists of horizontal component \( 2 \mathbf{i} \) and a vertical component \( 4 \mathbf{j} \).
02

Multiply Vector \(\mathbf{u}\) by 2

To find \(2 \mathbf{u}\), multiply each component of \(\mathbf{u}\) by 2. This gives us: \[ 2 \mathbf{u} = 2(3 \mathbf{i} - \mathbf{j}) = 6 \mathbf{i} - 2 \mathbf{j} \]
03

Subtract Vector \(\mathbf{v}\) from \(2 \mathbf{u}\)

Now, subtract the vector \(\mathbf{v}\) from the vector \(2 \mathbf{u}\): \[ 2 \mathbf{u} - \mathbf{v} = (6 \mathbf{i} - 2 \mathbf{j}) - (2 \mathbf{i} + 4 \mathbf{j}) \] When subtracting, subtract corresponding components: \[ = (6 - 2) \mathbf{i} + (-2 - 4) \mathbf{j} = 4 \mathbf{i} - 6 \mathbf{j} \]
04

Identify the Horizontal and Vertical Components

After performing the operations, the resulting vector is \(4 \mathbf{i} - 6 \mathbf{j}\). The horizontal component is the coefficient of \(\mathbf{i}\), which is \(4\). The vertical component is the coefficient of \(\mathbf{j}\), which is \(-6\).

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Key Concepts

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

Horizontal Component
A vector's horizontal component indicates its direction along the x-axis. It is usually represented by a term involving the unit vector \( \mathbf{i} \). In the exercise above, the horizontal component helps us understand how far the vector moves from side to side.

For the vector \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \), the horizontal component is \( 3 \mathbf{i} \). Similarly, the horizontal part of \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \) is \( 2 \mathbf{i} \).
  • In vector arithmetic, when vectors are scaled (like multiplying \( \mathbf{u} \) by 2), each component must be scaled accordingly. This keeps the direction and balance intact.
  • So, multiplying \( \mathbf{u} \) by 2 gives \( 6 \mathbf{i} \), representing the enhanced horizontal movement.
  • In the final computation of \( 2 \mathbf{u} - \mathbf{v} \), subtracting \( 2 \mathbf{i} \) from \( 6 \mathbf{i} \) results in \( 4 \mathbf{i} \), defining the net horizontal motion of the vector.
Vertical Component
The vertical component of a vector shows its movement along the y-axis, represented by the unit vector \( \mathbf{j} \). Think of it like how far a vector will move up or down.

For the initial vector \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \), the vertical component is \( -\mathbf{j} \), indicating a downward movement. In contrast, the vector \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \) has a vertical component of \( 4 \mathbf{j} \), suggesting upward movement.
  • When multiplying vector \( \mathbf{u} \) by 2, the vertical component becomes \( -2 \mathbf{j} \). This shifts the vector further down.
  • Subtracting \( 4 \mathbf{j} \) from \( -2 \mathbf{j} \) during subtraction results in \( -6 \mathbf{j} \), showing an overall downward trend in the result.
Vector Subtraction
Vector subtraction is a process of determining the difference between two vectors. It is similar to vector addition but involves subtracting corresponding components of each vector.

When subtracting \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \) from \( 2 \mathbf{u} \), you need to counterbalance each component individually: horizontal minus horizontal, and vertical minus vertical.
  • The operation \( 6 \mathbf{i} - 2 \mathbf{i} = 4 \mathbf{i} \) gives the net horizontal component.
  • Similarly, \( -2 \mathbf{j} - 4 \mathbf{j} = -6 \mathbf{j} \) result in the net vertical component.
  • This ensures both magnitude and direction are taken into account, reflecting a true change from one vector to another.
Subtracting vectors rearranges the original journey, telling us how much direction and distance changed from one vector to the resulting expression.

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Most popular questions from this chapter

Sketch the vectors \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}\) - \(\mathbf{v}\). $$ \mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \mathbf{v}=-\mathbf{i}+2 \mathbf{j} $$

Sketch the vectors \(2 \mathbf{v}\) and \(-2 \mathbf{v}\). $$ \mathbf{v}=\langle-2,1\rangle $$

In Problems \(49-62, \mathbf{u}=\langle 2,-3\rangle, \mathbf{v}=\langle-1,5\rangle,\) and \(\mathbf{w}=\langle 3,-2\rangle .\) Find the indicated scalar or vector. $$ (2 \mathbf{v}) \cdot(3 \mathbf{w}) $$

In Problems \(8_{3}\) and \(84, \mathbf{u}=4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{v}=-\mathbf{i}+\mathbf{j}\). Find the indicated vector. $$ \operatorname{proj}_{\mathbf{u}-\mathbf{v}} \mathbf{v} $$

Find \(\mathbf{u}-4 \mathbf{v}\) and \(2 \mathbf{u}+5 \mathbf{v}\). $$ \mathbf{u}=5 \mathbf{i}-10 \mathbf{j}, \mathbf{v}=-10 \mathbf{i} $$
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