Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

If \(\mathbf{v}=\langle a, b\rangle\) and \(k\) is any real number, prove that the magnitude of \(k \mathbf{v}\) is \(|k|\) times the magnitude of \(\mathbf{v}\).

Step-by-Step Solution

Answer

The magnitude of \( k\mathbf{v} \) is \( |k| \cdot ||\mathbf{v}|| \), showing it's \(|k|\) times the magnitude of \( \mathbf{v} \).

Step 1: Define the Magnitude of a Vector

The magnitude of a vector \( \mathbf{v} = \langle a, b \rangle \) is given by \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \). This formula comes from the Pythagorean theorem and calculates the "length" of the vector.

Step 2: Compute the Magnitude of k times v

If we multiply the vector \( \mathbf{v} \) by a scalar \( k \), the new vector is \( k\mathbf{v} = \langle ka, kb \rangle \). The magnitude of this new vector is \( ||k\mathbf{v}|| = \sqrt{(ka)^2 + (kb)^2} = \sqrt{k^2a^2 + k^2b^2} \).

Step 3: Simplifying the Magnitude Formula

From the previous expression, we can factor out \( k^2 \): \( ||k\mathbf{v}|| = \sqrt{k^2(a^2 + b^2)} = \sqrt{k^2} \cdot \sqrt{a^2 + b^2} = |k| \cdot \sqrt{a^2 + b^2} \). The absolute value is used because magnitude (length) cannot be negative.

Step 4: Compare to the Magnitude of Original Vector

Recall that the original vector \( \mathbf{v} = \langle a,b \rangle \) had a magnitude of \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \). By comparison, the magnitude of \( k\mathbf{v} \) is \( |k| \cdot ||\mathbf{v}|| \), proving that the magnitude of \( k\mathbf{v} \) is indeed \(|k|\) times the magnitude of \( \mathbf{v} \).

Key Concepts

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar, which is a single real number. This operation scales the vector by that scalar.
To understand scalar multiplication better, imagine a vector \( \mathbf{v} = \langle a, b \rangle \), representing a direction and magnitude in a two-dimensional plane.

• If this vector is multiplied by a scalar \( k \), the vector's length changes, but its direction remains the same unless \( k \) is negative.
• The result of multiplying a vector \( \mathbf{v} \) by a scalar \( k \) is a new vector: \( k\mathbf{v} = \langle ka, kb \rangle \).
• If \( k \) is positive, the new vector is in the same direction as \( \mathbf{v} \). If \( k \) is negative, the vector points in the opposite direction.

By multiplying each component of the vector by the scalar, the size of the vector changes accordingly, illustrating the concept of scaling.

Absolute Value

Absolute value might sound technical but is actually quite simple. An absolute value refers to a number's distance from zero on the number line, without considering whether it's positive or negative.

• The absolute value of any number \( x \), denoted \( |x| \), is always non-negative.
• For a positive number, the absolute value is the number itself: \( |x| = x \).
• For a negative number, the absolute value is the number's opposite: \( |x| = -x \).

In the context of vectors, when a vector \( \mathbf{v} \) is scaled by \( k \), its magnitude is affected by \( |k| \). This ensures the length remains a non-negative quantity, maintaining the geometric integrity of the vector's magnitude.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, used to relate the sides of a right-angled triangle. It states that the square of the hypotenuse (longest side) is the sum of the squares of the other two sides.
For a vector \( \mathbf{v} = \langle a, b \rangle \), the concept of the Pythagorean theorem helps calculate its magnitude.

• The magnitude is computed as \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \).
• This formula is derived from the Pythagorean theorem, treating the vector components as sides of a right triangle.
• Since the vector's magnitude represents the "length" in space, the Pythagorean theorem assures precision in this calculation.

By scaling a vector \( \mathbf{v} \) with a scalar \( k \), the theorem further guides the calculation of the new magnitude by \( ||k\mathbf{v}|| = |k| \cdot ||\mathbf{v}|| \), showcasing its crucial role in vector mathematics.