Problem 46 - Algebra and Trigonometry

Question

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}+\mathbf{w}\|\)

Step-by-Step Solution

Answer

\|\mathbf{v} + \mathbf{w}\| = \sqrt{5}\

Step 1: Add the Vectors

First, add vectors \( \mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j}\ \) and \( \mathbf{w} = -2 \mathbf{i} + 3 \mathbf{j}\ \).The result is \( \mathbf{v} + \mathbf{w} = (3 - 2) \mathbf{i} + (-5 + 3) \mathbf{j} = \mathbf{i} - 2 \mathbf{j}\ \).

Step 2: Use the Magnitude Formula

To find the magnitude of the resulting vector, use the formula \( \| \mathbf{a} \| = \sqrt{x^2 + y^2}\ \), where \( \mathbf{a} = x \mathbf{i} + y \mathbf{j} \).

Step 3: Substitute the Values

Substitute the values \( x = 1 \ \) and \( y = -2 \ \) to get \( \| \mathbf{v} + \mathbf{w} \| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \.\ \).

Key Concepts

Vector Magnitude

When working with vectors, it's essential to understand the concept of vector magnitude. The magnitude of a vector is a measure of its length, and it's always a non-negative number.
The formula for finding the magnitude of a vector \(\textbf{a} = x \mathbf{i} + y \mathbf{j}\) is given by: \[ \| \textbf{a} \| = \sqrt{x^2 + y^2} \] This formula is derived from the Pythagorean theorem. It essentially calculates the hypotenuse of a right triangle formed by the vector's components.
For example, if we have a vector \(\textbf{a} = \mathbf{i} - 2\mathbf{j}\), we can find its magnitude as follows: \[ \| \textbf{a} \| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \] So, the magnitude of the vector \(\textbf{a}\) is \(\sqrt{5}\). This length tells us how far the vector stretches from the origin to its endpoint.
Understanding vector magnitude is crucial in many practical applications, such as physics, engineering, and computer graphics.

Resultant Vector

The resultant vector is what you get when you add two or more vectors together. It represents the cumulative effect of combining these vectors.
To find the resultant vector, you add the individual components of the vectors. For instance, given vectors \(\textbf{v} = 3 \mathbf{i} - 5 \mathbf{j}\) and \(\textbf{w} = -2 \mathbf{i} + 3 \mathbf{j}\), the resultant vector \(\textbf{v} + \textbf{w}\) is calculated as follows: \[ \mathbf{v} + \mathbf{w} = (3 - 2) \mathbf{i} + (-5 + 3) \mathbf{j} = \mathbf{i} - 2 \mathbf{j} \] The new vector \(\mathbf{i} - 2 \mathbf{j}\) is the resultant vector. It combines the effects of vectors \(\textbf{v}\) and \(\textbf{w}\) into a single vector.
This resultant vector can then be used in further calculations or visual representations, giving a clear picture of the overall direction and magnitude resulting from the original vectors.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates to vector magnitude. It states that for a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Mathematically, it's expressed as: \[ c^2 = a^2 + b^2 \] In the context of vectors, this theorem helps us find the magnitude of a vector by treating its components as the sides of a right triangle. Given a vector \(\mathbf{a} = x \mathbf{i} + y \mathbf{j}\), we use the Pythagorean theorem to find its magnitude: \[ \| \mathbf{a} \| = \sqrt{x^2 + y^2} \] This method essentially calculates the hypotenuse (or length) of the right triangle formed by the horizontal and vertical components of the vector.
The Pythagorean theorem provides a simple but powerful way to analyze vectors and is foundational in many areas, including physics, engineering, and mathematics.