Problem 42 - Precalculus

Question

Find a unit vector in the direction of the given vector. $$\mathbf{v}=\langle 40,-9\rangle$$

Step-by-Step Solution

Answer

The unit vector is \( \left( \frac{40}{41}, \frac{-9}{41} \right) \).

Step 1: Find the Magnitude of the Vector

To find a unit vector in the direction of a given vector, we first need to determine the magnitude of the vector. For the vector \( \mathbf{v} = \langle 40, -9 \rangle \), the magnitude is calculated as follows: \[\|\mathbf{v}\| = \sqrt{40^2 + (-9)^2} = \sqrt{1600 + 81} = \sqrt{1681} = 41\]

Step 2: Divide Each Component by the Magnitude

A unit vector in the direction of a given vector is found by dividing each component of the vector by its magnitude. The components of \( \mathbf{v} = \langle 40, -9 \rangle \) are divided by \( 41 \), the magnitude we found:\[\text{Unit vector} = \left( \frac{40}{41}, \frac{-9}{41} \right)\]

Step 3: Write the Unit Vector

The vector obtained by dividing each of the original vector's components by its magnitude is the unit vector. So the unit vector in the direction of \( \mathbf{v} \) is: \[\mathbf{u} = \left( \frac{40}{41}, \frac{-9}{41} \right)\]

Key Concepts

Vector Magnitude

The magnitude of a vector is essentially the length of the vector. It's a measure that helps us understand how far the vector extends in space. Computing the magnitude is a vital step when working with vectors, particularly when you need to find a unit vector. A unit vector is a vector with a magnitude of 1, indicating the direction but not the size of the original vector.

To find the magnitude of a vector with components, such as \( \mathbf{v} = \langle 40, -9 \rangle \), we apply the Pythagorean Theorem. In this case, we compute it as:

• Square each component: \( 40^2 = 1600 \) and \( (-9)^2 = 81 \).
• Add the squares: \( 1600 + 81 = 1681 \).
• Take the square root of the sum: \( \sqrt{1681} = 41 \).

By following these steps, we calculate that the magnitude of vector \( \mathbf{v} \) is 41.

Direction of a Vector

The direction of a vector indicates where the vector is pointed in space. It's essentially where the vector is heading. To express this mathematically, particularly for unit vectors, we normalize the original vector. That means we scale the vector to have a magnitude of 1.

This step involves using the vector's magnitude we calculated. For a vector \( \mathbf{v} = \langle 40, -9 \rangle \), whose magnitude is 41, normalizing the vector entails dividing each component by 41.

Thus, the components of the unit vector are:

• For the x-component: \( \frac{40}{41} \)
• For the y-component: \( \frac{-9}{41} \)

These fractions represent the direction of the vector without incorporating the magnitude, effectively expressing how the vector points.

Components of a Vector

The components of a vector are the building blocks that define the vector in a coordinate system. In a two-dimensional space, a vector can be broken down into two components: one along the x-axis and the other along the y-axis. For example, the vector \( \mathbf{v} = \langle 40, -9 \rangle \) has:

• X-component: 40
• Y-component: -9

The components provide a directional structure, enabling you to perform calculations such as determining the vector's magnitude or the unit vector associated with it.

When you calculate a unit vector, you essentially transform these components to shrink or extend the vector to a unit size, while preserving its direction. This is crucial for applications that require directional accuracy without concern for distance or mass, such as determining path directions in navigation systems.