Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

If \(\mathbf{v}=\langle a, b\rangle\), prove that the magnitude of \(2 \mathbf{v}\) is twice the magnitude of \(\mathbf{v}\).

Step-by-Step Solution

Answer

The magnitude of \(2\mathbf{v}\) is twice that of \(\mathbf{v}\).

Step 1: Understanding Vectors

The vector \( \mathbf{v} = \langle a, b \rangle \) is given, where \( a \) and \( b \) are the components of the vector in a 2-dimensional plane.

Step 2: Magnitude Formula

The magnitude of a vector \( \mathbf{v} = \langle a, b \rangle \) is given by the formula \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \).

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

Using the formula, the magnitude of \( \mathbf{v} \) is \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \).

Step 4: Scale the Vector

When multiplying the vector \( \mathbf{v} \) by 2, the new vector is \( 2\mathbf{v} = \langle 2a, 2b \rangle \).

Step 5: Magnitude of Scaled Vector

The magnitude of the vector \( 2\mathbf{v} = \langle 2a, 2b \rangle \) is calculated as \( ||2\mathbf{v}|| = \sqrt{(2a)^2 + (2b)^2} = \sqrt{4a^2 + 4b^2} \).

Step 6: Simplify the Magnitude

Simplify the expression to find \( ||2\mathbf{v}|| = \sqrt{4(a^2 + b^2)} = \sqrt{4} \cdot \sqrt{a^2 + b^2} = 2\sqrt{a^2 + b^2} \).

Step 7: Compare with Double the Original Magnitude

Notice that \( 2\sqrt{a^2 + b^2} = 2 ||\mathbf{v}|| \), showing that the magnitude of \( 2\mathbf{v} \) is twice the magnitude of \( \mathbf{v} \).

Key Concepts

Vector Scaling

Vector scaling is a process by which we multiply a vector by a scalar, or a single number. By doing so, we either increase or decrease the vector's magnitude, depending on the scalar's value.
For example, if you have a vector \( \mathbf{v} = \langle a, b \rangle \) and multiply it by 2, you will get a new vector \( 2\mathbf{v} = \langle 2a, 2b \rangle \).
Here are some key aspects of vector scaling:

• **Direction Remains Unchanged:** The direction of the vector does not change, only its length does.
• **Scalar Multiplication:** If the scalar is greater than 1, the vector's magnitude increases. If it's between 0 and 1, the magnitude decreases.
• **Zero Scalar:** Multiplying a vector by zero results in the zero vector, which has no direction and a magnitude of zero.

This concept can be observed in real-world examples such as stretching or shrinking an arrow on a map.
By using vector scaling, we can easily adjust the size of vectors in various applications while preserving their original direction.

Magnitude Formula

The magnitude of a vector is a measure of its length. For a two-dimensional vector \( \mathbf{v} = \langle a, b \rangle \), the magnitude \( ||\mathbf{v}|| \) is calculated using the Pythagorean theorem.
The formula used is:

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

This takes the components \( a \) and \( b \) along the horizontal and vertical axes, squares them, adds the squares, and finally takes the square root of the sum.This process gives you the length of the vector from the origin to the point \( (a, b) \) on a coordinate plane.

• **Real-world analogy:** Finding the magnitude is similar to finding the distance of a straight line between two points.
• **Applications:** Used in physics to determine speed or force, where direction does not matter.

When we scale a vector, as with \( 2\mathbf{v} = \langle 2a, 2b \rangle \), the magnitude becomes twice that of the original vector, confirming the relationship demonstrated through the magnitude formula.

Two-dimensional Vectors

Two-dimensional vectors are essential in representing quantities with direction and magnitude in a plane.
These vectors have two components, usually written as \( \langle a, b \rangle \), where \( a \) and \( b \) correspond to measurements along the x and y axes, respectively.
Understanding two-dimensional vectors involves a few key points:

• **Components:** These form the building blocks for determining both the direction and magnitude of the vector.
• **Graphical Representation:** Can be visualized as arrows pointing from the origin to the point \( (a, b) \) on a graph.
• **Vector Arithmetic:** Vectors can be added or subtracted based on their components.
• **Utilization:** Used in various fields such as computer graphics, physics, and engineering to model real-world phenomena.

By mastering concepts like the magnitude and direction of two-dimensional vectors, you can solve complex problems in diverse disciplines.
They offer a way to simplify and manage data representing attributes such as velocity, acceleration, or force.