Problem 21 - Algebra and Trigonometry

Question

19-22 Find the component of \(\mathbf{u}\) along \(\mathbf{v}\). $$\mathbf{u}=7 \mathbf{i}-24 \mathbf{j}, \quad \mathbf{v}=\mathbf{j}$$

Step-by-Step Solution

Answer

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is \(-24\).

Step 1: Understand the Problem

We need to find the component of vector \( \mathbf{u} = 7 \mathbf{i} - 24 \mathbf{j} \) along vector \( \mathbf{v} = \mathbf{j} \). This involves projecting one vector onto another.

Step 2: Projection Formula

The formula for finding the component of vector \( \mathbf{u} \) along vector \( \mathbf{v} \) is given by \( \text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} \), where \( \mathbf{u} \cdot \mathbf{v} \) is the dot product and \( \|\mathbf{v}\| \) is the magnitude of \( \mathbf{v} \).

Step 3: Dot Product

Calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \). We have \( \mathbf{u} = 7 \mathbf{i} - 24 \mathbf{j} \) and \( \mathbf{v} = \mathbf{j} \), so the dot product is: \( \mathbf{u} \cdot \mathbf{v} = (7 \mathbf{i} - 24 \mathbf{j}) \cdot (\mathbf{j}) = -24 \).

Step 4: Magnitude of \( \mathbf{v} \)

Find the magnitude \( \|\mathbf{v}\| \). Since \( \mathbf{v} = \mathbf{j} \), \( \|\mathbf{v}\| = \sqrt{0^2 + 1^2} = 1 \).

Step 5: Calculate Component

With the dot product and the magnitude, substitute them into the component formula: \( \text{comp}_{\mathbf{v}} \mathbf{u} = \frac{-24}{1} = -24 \). This means the component of \( \mathbf{u} \) along \( \mathbf{v} \) is \(-24\).

Key Concepts

Understanding the Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This number is important because it tells us about the relationship between the two vectors.

• For vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), the dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
• If two vectors are perpendicular, their dot product is zero.
• Conversely, the dot product is maximized for parallel vectors.

For our exercise with vectors \( \mathbf{u} = 7 \mathbf{i} - 24 \mathbf{j} \) and \( \mathbf{v} = \mathbf{j} \), the dot product is simply \( -24 \). This value is crucial in finding the projection.

Magnitude of a Vector

The magnitude of a vector is essentially its length and is a key component in vector analysis. Calculating it involves using the Pythagorean theorem in a vector form.

• For a vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \), the magnitude is given by \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2} \).
• Magnitude is always positive and provides a sense of scale of the vector.
• This value is important when you need to normalize a vector or compute something like the projection.

In the example, vector \( \mathbf{v} = \mathbf{j} \) has a magnitude of 1, since it is a unit vector along the y-axis.

Component of a Vector

Finding the component of a vector involves determining how much of one vector lies in the direction of another. It is like asking, "How much of vector \( \mathbf{u} \) is pointing in the direction of vector \( \mathbf{v} \)?"

• This component is essentially a scalar value.
• The calculation involves the dot product and the magnitude of the vector you are projecting onto.
• The component is a more understandable number than just the dot product, as it directly relates to the lengths of the vectors.

In the problem, the component of \( \mathbf{u} \) along \( \mathbf{v} \) is found to be \( -24 \). This means \( \mathbf{u} \) has \( -24 \) units of \( \mathbf{v} \).

Using the Projection Formula

The projection formula is a tool used to translate the relationship between vectors into more practical terms. Projections help in simplifying complex vector calculations.

• The formula for the component (projection) of \( \mathbf{u} \) along \( \mathbf{v} \) is: \[ \text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} \].
• Here, \( \mathbf{u} \cdot \mathbf{v} \) is the dot product and \( \|\mathbf{v}\| \) is the magnitude of \( \mathbf{v} \).
• This formula essentially scales the resultant vector to the appropriate magnitude, dictated by \( \mathbf{v} \).

In the current exercise, using the projection formula, the component becomes \( -24 \), denoting how much \( \mathbf{u} \) follows \( \mathbf{v} \). This scalar value portrays the specific contribution of \( \mathbf{u} \) in the direction of \( \mathbf{v} \).