Problem 59 - Algebra and Trigonometry

Question

If \(\mathbf{v}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=x \mathbf{i}+3 \mathbf{j},\) find all numbers \(x\) for which \(\|\mathbf{v}+\mathbf{w}\|=5\)

Step-by-Step Solution

Answer

x = -2 + \sqrt{21} or x = -2 - \sqrt{21}

Step 1 - Add the vectors

Find the sum of the vectors \(\mathbf{v} \) and \(\mathbf{w} \). Given \(\mathbf{v} = 2\mathbf{i} - \mathbf{j} \) and \(\mathbf{w} = x \mathbf{i} + 3 \mathbf{j} \), add them component-wise:\(\mathbf{v} + \mathbf{w} = (2 + x)\mathbf{i} + (-1 + 3) \mathbf{j} = (2 + x)\mathbf{i} + 2\mathbf{j} \)

Step 2 - Calculate the magnitude of the resultant vector

The magnitude of a vector \(\mathbf{a}\mathbf{i} + \mathbf{b}\mathbf{j} \) is given by \(\| \mathbf{a}\mathbf{i} + \mathbf{b}\mathbf{j} \| = \sqrt{a^2 + b^2} \). Here, \(\| \mathbf{v} + \mathbf{w} \| = \sqrt{(2 + x)^2 + 2^2} \).

Step 3 - Set up the magnitude equation

We are given that the magnitude \(\| \mathbf{v} + \mathbf{w} \| = 5 \). Set up the equation: \(\sqrt{(2 + x)^2 + 2^2} = 5 \).

Step 4 - Solve for x

Square both sides of the equation to eliminate the square root: \((2 + x)^2 + 4 = 25 \).Simplify and solve for \( x \): \((2 + x)^2 = 21 \), \(2 + x = \pm \sqrt{21} \).Thus, \( x = -2 + \sqrt{21} \) or \( x = -2 - \sqrt{21} \).

Key Concepts

Vector Magnitude

In mathematics and physics, the magnitude of a vector represents its size or length. It gives a measure of how long the vector is and is always a non-negative quantity.
The magnitude of a vector \(\mathbf{a}\mathbf{i} + \mathbf{b}\mathbf{j} \) is calculated using the Pythagorean theorem. This involves squaring each of the components, summing them up, and then taking the square root of the result.
The formula to find the magnitude of a vector \(\mathbf{a}\mathbf{i} + \mathbf{b}\mathbf{j} \) is:
\[ \| \mathbf{a}\mathbf{i} + \mathbf{b}\mathbf{j} \| = \sqrt{a^2 + b^2} \]
For example, the magnitude of the vector \(\mathbf{v} + \mathbf{w} = (2 + x)\mathbf{i} + 2\mathbf{j} \) is calculated as:
\[ \| (2 + x)\mathbf{i} + 2\mathbf{j} \| = \sqrt{(2 + x)^2 + 2^2} \]
This gives us a way to determine the length of the resultant vector.

Resultant Vector

A resultant vector is obtained by adding two or more vectors together. It represents the combined effect of those vectors.
To find the resultant vector, you add corresponding components of the individual vectors together.
For example, given vectors \(\mathbf{v} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{w} = x\mathbf{i} + 3\mathbf{j}\), the resultant vector \(\mathbf{v} + \mathbf{w}\) is found by adding the \(\mathbf{i} \) and \(\mathbf{j} \) components separately:

• The \(\mathbf{i} \) components: \(2 + x \)
• The \(\mathbf{j}\) components: \( -1 + 3 = 2 \)

The resultant vector then is \(\mathbf{v} + \mathbf{w} = (2 + x)\mathbf{i} + 2\mathbf{j}\). This vector represents the combined effect of the two original vectors in the context of the problem.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value(s) of a variable that make an equation true.
In the context of our problem, we need to find the value of \(x\) such that the magnitude of the resultant vector is 5.
We start with the equation: \[ \sqrt{(2 + x)^2 + 2^2} = 5 \]
To solve for \(x\), we get rid of the square root by squaring both sides:
\[ (2 + x)^2 + 4 = 25 \]
Simplify this equation:
\[ (2 + x)^2 = 21 \]
Next, solve for the expression \(2 + x\): \[ 2 + x = \pm \sqrt{21} \]
This yields two possible values for \(x\):
1. \( x = -2 + \sqrt{21} \)
2. \( x = -2 - \sqrt{21} \)
Both solutions are valid and must be considered in the context of the problem. Thus, solving equations helps us derive the exact values needed to satisfy given conditions.