Question

In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=2 \mathbf{i}-\mathbf{j}$

   In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$.
$\mathbf{v}=2 \mathbf{i}-\mathbf{j}$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 54

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In this case, $\mathbf{v} = 2\mathbf{i} - \mathbf{j}$. Show more…

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In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=2 \mathbf{i}-\mathbf{j}$

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Key Concepts

Vector

A vector is a mathematical object that has both magnitude (length) and direction. It is typically represented in a coordinate system by its components along the axes. Vectors are fundamental in representing physical quantities such as force, velocity, and displacement, as they encapsulate both how much and in which direction the quantity acts.

Magnitude

The magnitude of a vector is a measure of its length in the vector space. It is calculated by taking the square root of the sum of the squares of its components. This scalar quantity is essential because it helps quantify the size of the vector independent of its direction, which is important for operations like normalization.

Normalization

Normalization is the process of converting a vector into a unit vector—one that has a magnitude of 1—while keeping its direction unchanged. This is typically accomplished by dividing each component of the vector by the vector's magnitude. Normalized vectors are useful in various applications, including defining directions and simplifying calculations in physics and geometry.

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