Problem 27 - Algebra and Trigonometry

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

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 \( \mathbf{v} = \mathbf{j} \). The component of a vector \( \mathbf{u} \) along another vector \( \mathbf{v} \) can be found using the dot product and the magnitude of \( \mathbf{v} \).

Step 2: Find the Dot Product \( \mathbf{u} \cdot \mathbf{v} \)

The dot product of two vectors \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) and \( \mathbf{v} = c \mathbf{i} + d \mathbf{j} \) is given by:\[ \mathbf{u} \cdot \mathbf{v} = ac + bd \]For our vectors, \( \mathbf{u} \cdot \mathbf{v} = 7 \cdot 0 + (-24) \cdot 1 = -24 \). Thus, \( \mathbf{u} \cdot \mathbf{v} = -24 \).

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

The magnitude of a vector \( \mathbf{v} = c \mathbf{i} + d \mathbf{j} \) is calculated as:\[ \| \mathbf{v} \| = \sqrt{c^2 + d^2} \]Since \( \mathbf{v} = \mathbf{j} \), \( c = 0 \) and \( d = 1 \). Thus, \( \| \mathbf{v} \| = \sqrt{0^2 + 1^2} = 1 \).

Step 4: Find the Component of \( \mathbf{u} \) along \( \mathbf{v} \)

The component of vector \( \mathbf{u} \) along vector \( \mathbf{v} \) is given by the formula:\[ \text{component of } \mathbf{u} \text{ along } \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{v} \|} \]Substitute the values:\[ \frac{-24}{1} = -24 \]

Step 5: Conclude the Solution

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is \(-24\), which is the measure of how much of \( \mathbf{u} \) points in the direction of \( \mathbf{v} \).

Key Concepts

Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar (a single number). It measures how much one vector extends in the direction of another. If you have two vectors, like \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their dot product is calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

This formula sums the products of the corresponding components.
For vectors \( \mathbf{u} = 7\mathbf{i} - 24\mathbf{j} \) and \( \mathbf{v} = \mathbf{j} \), the dot product becomes:\[ 7 \cdot 0 + (-24) \cdot 1 = -24 \]
This value, \(-24\), represents how aligned the vectors are. A negative result indicates opposite directions. This step is crucial for finding how much of one vector is along another.

Magnitude of a Vector

The magnitude of a vector is its length or size. It tells you how long the vector is in space, regardless of its direction. To find the magnitude, we use the Pythagorean theorem, writing the formula as:

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

Here, a vector \( \mathbf{v} \) is expressed in terms of its components \(c\) and \(d\).
For example, the vector \( \mathbf{v} = \mathbf{j} \) has components \(c = 0\) and \(d = 1\).
The calculation of its magnitude will be:\[ \| \mathbf{v} \| = \sqrt{0^2 + 1^2} = 1 \]
This result means the vector \(\mathbf{v}\) is one unit long. Understanding magnitude helps in normalizing vectors and finding projections.

Component of a Vector

The component of one vector along another gives a measure of how much of the first vector lies in the direction of the second vector. It's derived from the dot product and the magnitude.
The formula to calculate this component is:

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

This formula divides the dot product by the magnitude of the vector \( \mathbf{v} \).
For instance, if \( \mathbf{u} \cdot \mathbf{v} = -24 \) and \( \| \mathbf{v} \| = 1 \), then:

• \[ \frac{-24}{1} = -24 \]

This means that the vector \( \mathbf{u} \) extends \(-24\) units along the direction of \( \mathbf{v} \).
The negative sign indicates the direction is opposite.