Problem 27 - Precalculus Mathematics for Calculus

Question

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

-24

Step 1: Calculate the Dot Product

First, we need to compute the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \). The dot product is given by the formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). In this case, \( \mathbf{u} = 7\mathbf{i} - 24\mathbf{j} \) and \( \mathbf{v} = \mathbf{j} = 0\mathbf{i} + 1\mathbf{j} \), so: \[ \mathbf{u} \cdot \mathbf{v} = 7 \times 0 + (-24) \times 1 = -24. \]

Step 2: Compute the Magnitude of \(\mathbf{v}\)

To find the component, we also need the magnitude of \( \mathbf{v} \). The magnitude of \( \mathbf{v} \) is calculated as: \[ ||\mathbf{v}|| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1. \]

Step 3: Calculate the Component Along \(\mathbf{v}\)

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is given by the formula: \[ \text{component of } \mathbf{u} \text{ along } \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{v}||}. \] Substituting in the values from previous steps, we find: \[ \frac{-24}{1} = -24. \]

Key Concepts

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single number, or scalar. It is a fundamental concept in vector algebra. To compute the dot product of two vectors, you multiply corresponding components and then sum them up. It's represented by the formula:

• If two vectors are in the form \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), then the dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]

This operation can tell us about the angle between the two vectors. If the dot product is zero, the vectors are perpendicular. In our exercise, the dot product of \( \mathbf{u} = 7 \mathbf{i} - 24 \mathbf{j} \) and \( \mathbf{v} = \mathbf{j} \) results in -24. The negative result indicates that there is an obtuse angle between the two vectors.

Vector Magnitude

The magnitude of a vector, also known as its length or norm, is a measure of how long it is. For a 2D vector, it can be visualized as the length of the line segment from the origin to the point represented by the vector. The magnitude of a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \) is given by the formula:

• \[ ||\mathbf{v}|| = \sqrt{a^2 + b^2} \]

In our particular exercise, vector \( \mathbf{v} \) is just \( \mathbf{j} \), which is a unit vector along the y-axis with a magnitude of 1. This means that any component of another vector along \( \mathbf{v} \) will not change its magnitude, essentially maintaining the integrity of whatever is being measured along it.

Projection of a Vector

The projection of a vector \( \mathbf{u} \) onto another vector \( \mathbf{v} \) is a way of finding a component of \( \mathbf{u} \) that points in the same direction as \( \mathbf{v} \). It is as if we are squeezing the vector \( \mathbf{u} \) onto the line of \( \mathbf{v} \). The length of this projection is given by the formula:

• \(\text{Component of } \mathbf{u} \text{ along } \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{v}||}\)

In our exercise, following the calculations, the component of \( \mathbf{u} \) along \( \mathbf{v} \) was found to be -24. This means \( \mathbf{u} \) contributes -24 units in the direction of \( \mathbf{v} \). Such computations are crucial in physics and engineering, where we often need to find how much of a vector quantity (like force or velocity) acts in a specific direction.