Problem 38 - Algebra and Trigonometry

Question

In Exercises \(21-38,\) let $$\mathbf{u}=2 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}, \text { and } \mathbf{w}=-\mathbf{i}-6 \mathbf{j}$$ Find each specified vector or scalar. $$|\mathbf{u}-\mathbf{w}|$$

Step-by-Step Solution

Answer

The magnitude of the vector difference between \(\mathbf{u}\) and \(\mathbf{w}\) is \(\sqrt{10}\)

Step 1: Subtract vectors

To find \(\mathbf{u}-\mathbf{w}\), subtract the corresponding components of \(\mathbf{w}\) from \(\mathbf{u}\). So, \[\mathbf{u}-\mathbf{w}= (2-(-1))\mathbf{i}+(-5-(-6))\mathbf{j} = 3\mathbf{i}-(-1)\mathbf{j} = 3\mathbf{i}+\mathbf{j}.\]

Step 2: Calculate the magnitude

The magnitude of a vector \(\mathbf{v} = a\mathbf{i}+b\mathbf{j}\) is given by \(\sqrt{a^2 + b^2}\). So, using this formula the magnitude of \(\mathbf{u}-\mathbf{w}\) is \[\sqrt{(3)^2 + (1)^2} = \sqrt{10}.\]

Key Concepts

Magnitude of a Vector

Understanding the magnitude of a vector is crucial in vector arithmetic. It represents the length or size of the vector and is always a non-negative number. Consider a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \).
The magnitude, denoted as \(|\mathbf{v}|\), is calculated using the formula: \[|\mathbf{v}| = \sqrt{a^2 + b^2}. \]This comes from the Pythagorean theorem, which shows us the distance from the origin to the point \((a, b)\) in the coordinate plane.
It's similar to finding the hypotenuse of a right triangle where \( a \) and \( b \) are the legs. For example, if you subtract vectors as in \( \mathbf{u}-\mathbf{w} = 3\mathbf{i} + \mathbf{j} \), you find the magnitude by evaluating: \[\sqrt{3^2 + 1^2} = \sqrt{10}.
\] This gives you a scalar value representing how long the resultant vector is.

Components of a Vector

Vectors are defined by their components, which effectively tell us the direction and magnitude in each axis. When dealing with two-dimensional vectors, components are usually expressed in terms of \( \mathbf{i} \) and \( \mathbf{j} \), which correspond to the horizontal and vertical axes.
For any vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \), \( a \) is the coefficient of \( \mathbf{i} \) that represents the horizontal component. Similarly, \( b \) is the coefficient of \( \mathbf{j} \) representing the vertical component.
When subtracting vectors like \( \mathbf{u} - \mathbf{w} \), you simply subtract each respective component:

• Horizontal: \( 2 - (-1) = 3 \)
• Vertical: \( -5 - (-6) = 1 \)

Such calculations give you the resultant vector: \( 3\mathbf{i} + \mathbf{j} \). Components play a vital role in understanding how vectors combine and interact in space.

Vector Arithmetic

Vector arithmetic allows us to perform operations like addition and subtraction with vectors, allowing us to solve complex mathematical and physics problems.
Subtracting vectors involves element-wise subtraction of their components. For vectors \( \mathbf{a} \) and \( \mathbf{b} \), the vector subtraction \( \mathbf{a} - \mathbf{b} \) is calculated as:

• Subtracting \( \mathbf{i} \) components: \( a_i - b_i \)
• Subtracting \( \mathbf{j} \) components: \( a_j - b_j \)

In the given exercise, this operation appears as \( \mathbf{u}-\mathbf{w} = 3\mathbf{i} + \mathbf{j} \).
This equation results from straightforward arithmetic operations that ensure both the direction and magnitude of the resulting vector are captured.
Understanding these basic operations forms the foundation for more complex vector algebra applications in geometry and physics.