Problem 28 - Precalculus Mathematics for Calculus

Question

Find the component of \(\mathbf{u}\) along \(\mathbf{v}\). $$\mathbf{u}=7 \mathbf{i}, \quad \mathbf{v}=8 \mathbf{i}+6 \mathbf{j}$$

Step-by-Step Solution

Answer

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is 5.6.

Step 1: Understand the Dot Product Formula

To find the component of \( \mathbf{u} \) along \( \mathbf{v} \), we utilize the formula for the projection of \( \mathbf{u} \) onto \( \mathbf{v} \), which is given by \( \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} \). Therefore, we need to calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \) and the magnitude squared of \( \mathbf{v} \), which is \( \|\mathbf{v}\|^2 \).

Step 2: Calculate the Dot Product

The dot product of \( \mathbf{u} = 7 \mathbf{i} \) and \( \mathbf{v} = 8 \mathbf{i} + 6 \mathbf{j} \) is calculated as follows: \( \mathbf{u} \cdot \mathbf{v} = (7 \mathbf{i}) \cdot (8 \mathbf{i} + 6 \mathbf{j}) = 7 \cdot 8 + 0 \cdot 6 = 56 \).

Step 3: Compute the Magnitude Squared of \( \mathbf{v} \)

Compute \( \|\mathbf{v}\|^2 \) by calculating the square of the magnitudes of its components: \( \mathbf{v} = 8 \mathbf{i} + 6 \mathbf{j} \), hence \( \|\mathbf{v}\|^2 = 8^2 + 6^2 = 64 + 36 = 100 \).

Step 4: Find the Projection Scalar

Now, find the component (or the scalar of the projection) of \( \mathbf{u} \) along \( \mathbf{v} \) using the formula: \( \text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} = \frac{56}{\sqrt{100}} \).

Step 5: Simplify

Simplify the expression to find the projection scalar: \( \frac{56}{\sqrt{100}} = \frac{56}{10} = 5.6 \).

Key Concepts

Dot Product

The dot product is a way to multiply two vectors, producing a scalar (a single number) as an output. It's very useful in various geometric and physical calculations.
The formula for the dot product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \)

For vector \( \mathbf{u} \) and \( \mathbf{v} \), the dot product is calculated by multiplying their corresponding components and summing the results.
This forms a fundamental part of finding the projection of one vector onto another. It is important to know that the dot product involves multiplying the magnitudes of these vectors and the cosine of the angle between them.

Magnitude of a Vector

The magnitude of a vector, often depicted as \( \|\mathbf{v}\| \), essentially tells us the length or size of the vector. Visualize it as the distance from the origin to the point described by the vector. For a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \), its magnitude is calculated using the Pythagorean theorem:

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

This computation measures how "strong" or "far" a vector is in terms of its direction and provides the necessary scaling factor when projecting one vector onto another.
For example, when calculating the projection of one vector onto another, we need to establish the magnitude squared to properly scale the dot product result and derive accurate results.

Projection of a Vector

When we talk about the projection of a vector, we are looking to express one vector in terms of another vector's direction. It is like shining a shadow of one vector onto another. The formula for the projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is:

• \( \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \)

This tells us how much of \( \mathbf{u} \) is stretched along the line of \( \mathbf{v} \).
If a vector \( \mathbf{u} \) lies completely along the direction of \( \mathbf{v} \), then the projection of \( \mathbf{u} \) onto \( \mathbf{v} \) would perfectly match \( \mathbf{u} \). However, if \( \mathbf{u} \) is perpendicular to \( \mathbf{v} \), the projection would be zero, because there is no component of \( \mathbf{u} \) along \( \mathbf{v} \).

Component of a Vector

The component of a vector refers to how much of one vector is aligned with another vector. It is found through the projection process and can be thought of as a scalar that zooms in on certain aspects of the vector when combined with other vectors.
In mathematical terms, the component of \( \mathbf{u} \) along \( \mathbf{v} \) is given by:

• \( \text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} \)

This component is particularly useful, for example, in physics, when looking to understand how forces decompose into various directions.
It captures the essence of how the vector \( \mathbf{u} \) interacts or aligns along the vector \( \mathbf{v} \), enabling a deeper understanding of vector geometry and dynamics.