Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the dot product to find the magnitude of \(\mathbf{u}.\) $$\mathbf{u}=\langle-5,12\rangle$$

Step-by-Step Solution

Answer

The magnitude of the vector \(\mathbf{u}=\langle-5,12\rangle\) is \( ||\mathbf{u}|| = 13 \).

Step 1: Identify the components of the vector

The given vector \(\mathbf{u}\) has two components: -5 and 12.

Step 2: Apply the dot product formula for finding magnitude

The dot product of vector \(\mathbf{u}\) with itself is equal to the magnitude of \(\mathbf{u}\) squared. So, we calculate \( \mathbf{u} \cdot \mathbf{u} = (-5)^2 + (12)^2 \).

Step 3: Evaluate the expression

Evaluate the expression to get \( \mathbf{u} \cdot \mathbf{u} = 25 + 144 = 169 \).

Step 4: Calculate the magnitude

The magnitude of vector \(\mathbf{u}\) is the square root of the dot product of \(\mathbf{u}\) with itself, so \( ||\mathbf{u}|| = \sqrt{169} \).

Key Concepts

Vector Magnitude

The magnitude of a vector gives us an idea of how long the vector is. Think of it as the length of an arrow in space. For a vector \( \mathbf{u} \) with components \( \langle a, b \rangle \), the magnitude is calculated using the formula:

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

This formula essentially comes from the Pythagorean theorem, which applies to the right triangle formed by the vector components on a coordinate plane.
For example, if a vector has components \(-5\) and \(12\), its magnitude is found by computing \( \sqrt{(-5)^2 + (12)^2} \),
which results in \( \sqrt{169} = 13 \). Thus, the vector’s length is 13 units.

Vector Components

Vector components are the building blocks of a vector. They show how far the vector moves in each direction. A 2D vector \( \mathbf{u} = \langle a, b \rangle \) consists of:

• The horizontal movement, \( a \)
• The vertical movement, \( b \)

These components help us understand where the vector points and how it's oriented. When given a vector like \( \mathbf{u} = \langle -5, 12 \rangle \),
the components \(-5\) and \(12\) tell us it moves 5 units left along the x-axis and 12 units up along the y-axis.
This is important when working with vectors because we often need to add or subtract vectors, requiring us to know their components clearly.

Dot Product Formula

The dot product is a way of multiplying two vectors to get a scalar (a single number).
For vectors \( \mathbf{u} = \langle a, b \rangle \) and \( \mathbf{v} = \langle c, d \rangle \), their dot product is:

• \( \mathbf{u} \cdot \mathbf{v} = ac + bd \)

This is useful for various applications, such as finding the angle between vectors or determining orthogonality.
In the given problem, we use the dot product to find the magnitude of a vector by calculating \( \mathbf{u} \cdot \mathbf{u} = a^2 + b^2 \).
This gives us the squared magnitude of the vector, which can then be used to easily find the actual magnitude by taking the square root.
For \( \mathbf{u} = \langle -5, 12 \rangle \),
the dot product \( \mathbf{u} \cdot \mathbf{u} = (-5)^2 + (12)^2 = 25 + 144 = 169 \),
leading to the magnitude \( ||\mathbf{u}|| = \sqrt{169} = 13 \).