Problem 45 - Precalculus with Limits

Question

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{w} = 4\mathbf{j}\)

Step-by-Step Solution

Answer

The unit vector in the direction of vector \(\mathbf{w}\) is \(\mathbf{u} = \mathbf{j}\). This has been verified as having a magnitude of 1.

Step 1: Calculate the Magnitude of the given vector

The vector \(\mathbf{w}\) only contains a j-component, therefore the magnitude is just the absolute of that component:\[|\mathbf{w}| = |4| = 4\]

Step 2: Calculate the unit vector

Now to find the unit vector in the direction of the given vector, divide the vector by its magnitude:\[\mathbf{u} = \frac{\mathbf{w}}{|\mathbf{w}|} = \frac{4\mathbf{j}}{4} = \mathbf{j}\]

Step 3: Verify the magnitude of the unit vector

It is stated in the exercise to verify that the magnitude of the result is indeed 1:\[|\mathbf{u}| = |\mathbf{j}| = 1\]The result confirms that the magnitude of the unit vector is 1, which means that the calculations are correct.

Key Concepts

Magnitude of a Vector

The magnitude of a vector is a crucial concept in understanding the size or length of a vector. When you visualize a vector, you can think of it as an arrow pointing from one point to another in a space. The magnitude is like measuring the length of this arrow. Mathematically, it's a scalar quantity, which means it has no direction.

To find the magnitude, use the formula based on the vector's components. For a vector \(\mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}\), the magnitude \( |\mathbf{v}|\) is calculated as:

• \(|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}\)

If the vector is two-dimensional, like \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\), the formula simplifies:

• \(|\mathbf{v}| = \sqrt{a^2 + b^2}\)

For the vector \(\mathbf{w} = 4\mathbf{j}\) in the original problem, it only has a j-component, making the calculation even simpler:

• \(|\mathbf{w}| = \sqrt{0^2 + 4^2} = 4\)

The magnitude tells us how long the vector is, regardless of its direction.

Direction of a Vector

The direction of a vector shows where the vector is pointing. Think of it like the direction an arrow is aimed. A vector’s direction is essential because, along with its magnitude, it fully describes the vector.

For a unit vector, the direction remains the same as the original vector, but its magnitude becomes 1. This is because each component of the vector is scaled down appropriately.

To determine the direction, especially in two dimensions, you can use the tangent of the angle formed between the vector and the horizontal axis:

• \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\),\ where \( a\) and \( b\) are the components of the vector.

For our vector \(\mathbf{w} = 4\mathbf{j}\), it lies entirely along the y-axis with no x-component, making its direction straightforward:

• It points directly upwards along the y-axis.

Thus, for this exercise's vector, the direction is vertical and aligns with the positive y-axis.

Vector Components

Vector components break a vector down into its fundamental parts. In the Cartesian coordinate system, a vector is a sum of components along the axes. This is similar to describing how far along each axis the vector travels.

Take a common two-dimensional vector \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\). Here, \(a\mathbf{i}\) is the component along the x-axis, and \(b\mathbf{j}\) is the component along the y-axis. For three dimensions, the vector can also have a component \(c\mathbf{k}\) along the z-axis.

• X-component: \(a\mathbf{i}\)
• Y-component: \(b\mathbf{j}\)
• Z-component: \(c\mathbf{k}\)

In our specific exercise, \(\mathbf{w} = 4\mathbf{j}\) indicates it only has a component along the y-axis. This means the vector's entire influence is directed along the vertical axis, and there's no movement in the horizontal plane.

Understanding components helps in visualizing where the vector is acting in space, making it easier to predict the resultant behaviors when vectors are added or subtracted. Each component represents a piece of the total vector, and together, they build the complete picture.