In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = -5j

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<insert step 1> Identify the given vector \( \mathbf{v} = -5\mathbf{j} \). This vector is in the direction of the negative y-axis.>

<insert step 2> Calculate the magnitude of the vector \( \mathbf{v} \). The magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is given by \( \|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2} \). For \( \mathbf{v} = -5\mathbf{j} \), \( a = 0, b = -5, c = 0 \).>

<insert step 3> Substitute the values into the magnitude formula: \( \|\mathbf{v}\| = \sqrt{0^2 + (-5)^2 + 0^2} = \sqrt{25} \).>

<insert step 4> Simplify the expression to find the magnitude: \( \|\mathbf{v}\| = 5 \).>

<insert step 5> Find the unit vector \( \mathbf{u} \) by dividing the vector \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-5\mathbf{j}}{5} = -\mathbf{j} \).>

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates direction. To find a unit vector in the same direction as a given vector, you divide the vector by its magnitude. This process normalizes the vector, allowing it to retain its direction while having a standardized length.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y² + z²) for three-dimensional vectors. For a two-dimensional vector like v = -5j, the magnitude is simply the absolute value of its components. Understanding how to compute the magnitude is essential for normalizing the vector to create a unit vector.

Direction of a Vector

The direction of a vector indicates the path along which it acts and is often represented by the angle it makes with a reference axis. In the case of the vector v = -5j, it points directly downward along the y-axis. Recognizing the direction is crucial when finding a unit vector, as it ensures that the resulting unit vector maintains the same orientation.