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Chapter 3: Problem 56

\(\mathbf{v}=-5 \mathbf{i}+4 \mathbf{j}\)

Short Answer

Expert verified
The x-component of the vector is -5 and the y-component is 4. The vector can be represented as (-5, 4).

Step by step solution

01

Identify Vector Components

The given vector \(\mathbf{v}=-5 \mathbf{i}+4 \mathbf{j}\) is in component form. Here, -5 is the x-component (i-component) and 4 is the y-component (j-component). We denote it as (-5, 4).

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Key Concepts

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

i-component
The i-component refers to the horizontal component of a vector in a two-dimensional coordinate system. In the vector \[\mathbf{v} = -5 \mathbf{i} + 4 \mathbf{j},\]\(-5 \mathbf{i}\) represents the i-component. This component points in the direction of the positive or negative x-axis. Here, the coefficient **-5** indicates the magnitude and direction along the horizontal axis. The negative sign shows that it points in the opposite direction from the positive x-axis.
  • The word "direction" is key. Vectors are directional quantities.
  • The horizontal part of a vector is always aligned along the x-axis.
In practical terms, anytime you look at a component that is multiplied by \( \mathbf{i} \), you are examining how much the vector stretches along the horizontal axis.
j-component
The j-component is the vertical component of a vector in a two-dimensional space. In the expression \[-5 \mathbf{i} + 4 \mathbf{j},\]\(4 \mathbf{j}\) signifies the j-component. This component runs parallel to the y-axis.**4** is the coefficient and describes how far the vector extends on the vertical axis. Since it is positive, \(4 \mathbf{j}\) shows movement in the positive y-direction.
  • Understanding the sign of the j-component helps determine the vertical direction.
  • Vectors with positive j-components move upwards, while those with negative ones move downwards.
The j-component lets you visualize how a vector travels vertically in a coordinate plane.
component form
The component form of a vector offers a straightforward way to express and analyze vectors in terms of their individual horizontal and vertical elements. When we say the vector is in "component form," we mean it's expressed like this:\[\mathbf{v} = (-5, 4)\]Here, the components are directly extracted from the vector notation \[-5 \mathbf{i} + 4 \mathbf{j}.\]
  • The first value, -5, is the i-component (x-component).
  • The second value, 4, is the j-component (y-component).
This format is beneficial because it lets you immediately see the magnitude and direction along each axis, simplifying calculations like vector addition or subtraction. It also provides a clear picture of where the vector is heading in a plane, serving as a foundational step to understanding complex vector operations.

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