Problem 87 - Precalculus

Question

Explain the mistake that is made. Find the magnitude of the vector \langle-2,-8\rangle Solution: Factor the -1 \(-(2,8)\) Find the magnitude \(|\langle 2,8\rangle|=\sqrt{2^{2}+8^{2}}\) of \langle 2,8\rangle \(=\sqrt{68}=2 \sqrt{17}\) Write the magnitude of \langle-2,-8\rangle \(|\langle-2,-8\rangle|=-2 \sqrt{17}\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was assuming a negative magnitude; magnitude is always positive.

Step 1: Identify Original Vector Components

The given vector is \( \langle -2, -8 \rangle \). These are the components that define the vector's direction and magnitude in a 2D space.

Step 2: Understanding Magnitude Formula

The magnitude of a vector \( \langle a, b \rangle \) is calculated using the formula: \[ |\langle a, b \rangle| = \sqrt{a^2 + b^2} \] This is derived from the Pythagorean theorem to find the length of the vector from the origin to the point \( (a, b) \).

Step 3: Calculate Magnitude of Original Vector

Apply the magnitude formula to the original components: \[ |\langle -2, -8 \rangle| = \sqrt{(-2)^2 + (-8)^2} \] This simplifies to: \[ |\langle -2, -8 \rangle| = \sqrt{4 + 64} = \sqrt{68} = 2 \sqrt{17} \].

Step 4: Explanation of Mistake

The mistake was to assume that factoring out a negative sign from the vector components affects the magnitude's sign. Magnitude is always positive as it represents a length. Factoring \( -1 \) and calculating the magnitude of the resulting positive vector incorrectly led to the conclusion of a negative magnitude: \( |-2 \sqrt{17}| = 2 \sqrt{17} \). The final result should still be positive.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that helps us find the length of the hypotenuse in a right-angled triangle. Imagine you have a right triangle where the two shorter sides are called "a" and "b." To find the length of the hypotenuse, which is the longest side opposite the right angle, you use the formula: \[ c = \sqrt{a^2 + b^2} \] This theorem also plays a crucial role in understanding vectors, especially in two-dimensional space.

A vector can be thought of as represented by a right triangle, where its components, often referred to as "a" and "b," form the perpendicular sides. The vector itself represents the hypotenuse, usually denoted by "c." Therefore, to find a vector's magnitude—its length—we apply the Pythagorean theorem. This establishes a deep link between geometry and algebraic representation of vectors, revealing their inherent lengths and directions in a simple, calculable form.

vector components

Vector components are essential in breaking down a vector into simpler parts. A vector in two-dimensional space is represented as \( \langle a, b \rangle \), where "a" and "b" are its components. These components tell us about the direction and magnitude along the x-axis and y-axis, respectively.

Each component acts like the base and height of a right triangle originating from a common point, often the origin in a coordinate system. By thinking of these components, you can visualize how far and in which direction the vector stretches.

• "a" is how far left or right the vector moves.
• "b" is how far up or down the vector moves.

Understanding these individual parts helps calculate the vector's magnitude using the Pythagorean theorem, giving the full picture of how vectors behave and interact in mathematical spaces.

positive magnitude

Magnitude, in the context of vectors, always represents a length and is fundamentally a positive quantity. Imagine measuring distance with a ruler: you wouldn't have a negative distance! The magnitude of a vector, like \( \langle -2, -8 \rangle \), is calculated by squaring each component, summing them, and then taking the square root, which ensures the result is positive:\[ |\langle -2, -8 \rangle| = \sqrt{(-2)^2 + (-8)^2} = \sqrt{4 + 64} = \sqrt{68} = 2 \sqrt{17} \]

In the original solution, a mistake was made by considering the magnitude could be negative due to a negative factor (-1 in this case) wrongly influencing the calculations. Magnitude reflects the length of the vector, disregarding direction, embodying only the concept of size. Therefore, it remains unaffected by swapping the sign of vector components, ensuring it is always non-negative.