Problem 102 - Precalculus

Question

$$\text { Let } \mathbf{u}=\langle 2 a, a\rangle, \mathbf{v}=\langle-a,-2 a\rangle . \text { Compute }\left|\frac{2 \mathbf{u}}{|\mathbf{v}|}-\frac{3 \mathbf{v}}{|\mathbf{u}|}\right|$$

Step-by-Step Solution

Answer

The magnitude is \( \sqrt{\frac{113}{5}} \).

Step 1: Identify the Vectors

We have two vectors \( \mathbf{u} = \langle 2a, a \rangle \) and \( \mathbf{v} = \langle -a, -2a \rangle \). Our task is to manipulate these vectors according to the problem statement.

Step 2: Compute the Magnitude of \(\mathbf{v}\)

The magnitude of \( \mathbf{v} = \langle -a, -2a \rangle \) is calculated as: \[ |\mathbf{v}| = \sqrt{(-a)^2 + (-2a)^2} = \sqrt{a^2 + 4a^2} = \sqrt{5a^2} = a\sqrt{5}. \]

Step 3: Compute the Magnitude of \(\mathbf{u}\)

The magnitude of \( \mathbf{u} = \langle 2a, a \rangle \) is calculated similarly: \[ |\mathbf{u}| = \sqrt{(2a)^2 + a^2} = \sqrt{4a^2 + a^2} = \sqrt{5a^2} = a\sqrt{5}. \]

Step 4: Find \( 2\mathbf{u} \) and \( 3\mathbf{v} \)

Multiply the vectors by their respective scalars: \( 2\mathbf{u} = \langle 4a, 2a \rangle \) and \( 3\mathbf{v} = \langle -3a, -6a \rangle \).

Step 5: Compute \( \frac{2\mathbf{u}}{|\mathbf{v}|} \) and \( \frac{3\mathbf{v}}{|\mathbf{u}|} \)

Divide the vectors by the magnitudes:\[ \frac{2\mathbf{u}}{|\mathbf{v}|} = \frac{\langle 4a, 2a \rangle}{a\sqrt{5}} = \left\langle \frac{4a}{a\sqrt{5}}, \frac{2a}{a\sqrt{5}} \right\rangle = \left\langle \frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right\rangle, \]\[ \frac{3\mathbf{v}}{|\mathbf{u}|} = \frac{\langle -3a, -6a \rangle}{a\sqrt{5}} = \left\langle \frac{-3a}{a\sqrt{5}}, \frac{-6a}{a\sqrt{5}} \right\rangle = \left\langle \frac{-3}{\sqrt{5}}, \frac{-6}{\sqrt{5}} \right\rangle. \]

Step 6: Subtract the Two Vectors

Subtract \( \frac{3\mathbf{v}}{|\mathbf{u}|} \) from \( \frac{2\mathbf{u}}{|\mathbf{v}|} \):\[ \left\langle \frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right\rangle - \left\langle \frac{-3}{\sqrt{5}}, \frac{-6}{\sqrt{5}} \right\rangle = \left\langle \frac{4}{\sqrt{5}} + \frac{3}{\sqrt{5}}, \frac{2}{\sqrt{5}} + \frac{6}{\sqrt{5}} \right\rangle = \left\langle \frac{7}{\sqrt{5}}, \frac{8}{\sqrt{5}} \right\rangle. \]

Step 7: Compute the Magnitude of the Result

Finally, find the magnitude of the resultant vector:\[ \left| \frac{2 \mathbf{u}}{|\mathbf{v}|} - \frac{3 \mathbf{v}}{|\mathbf{u}|} \right| = \sqrt{\left( \frac{7}{\sqrt{5}} \right)^2 + \left( \frac{8}{\sqrt{5}} \right)^2} = \sqrt{\frac{49}{5} + \frac{64}{5}} = \sqrt{\frac{113}{5}} = \frac{\sqrt{113}}{\sqrt{5}}. \]

Step 8: Simplify the Result

Simplify the final expression for clarity:\[ \frac{\sqrt{113}}{\sqrt{5}} = \sqrt{\frac{113}{5}}. \]

Key Concepts

Magnitude of a Vector

In vector mathematics, the magnitude of a vector is essentially its length or size. This is calculated using the Pythagorean Theorem to sum up the squares of each component of the vector. Think of it as the distance from the origin in a coordinate system to the point specified by the vector.
For example, if you have a vector \( \mathbf{u} = \langle 2a, a \rangle \), you find its magnitude by calculating:

• Square each component: \((2a)^2 + (a)^2\)
• Add the squares: \(4a^2 + a^2 = 5a^2\)
• Take the square root: \(\sqrt{5a^2} = a\sqrt{5}\)

Hence, the magnitude is expressed as \( a\sqrt{5} \). The magnitude helps to understand the size of movement a vector can represent in its space.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is just a fancy word for a simple number. This operation scales the vector's magnitude but does not change its direction.
If you take the vector \( \mathbf{u} = \langle 2a, a \rangle \) and multiply it by a scalar 2, the result is \( 2\mathbf{u} = \langle 4a, 2a \rangle \). Here:

• Each component of the vector is multiplied by the scalar: \( 2 \times 2a \) and \( 2 \times a \)
• The resulting vector \( \langle 4a, 2a \rangle \) is in the same direction as the original vector

This operation enhances the magnitude of the original vector without altering its path, which can be represented visually as stretching or compressing the vector along the same line.

Vector Subtraction

Vector subtraction is an operation where one vector is subtracted from another. This involves subtracting their corresponding components to produce a new vector.
When subtracting vectors, like in the exercise where you have \( \frac{2\mathbf{u}}{|\mathbf{v}|} = \langle \frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}} \rangle \) and \( \frac{3\mathbf{v}}{|\mathbf{u}|} = \langle \frac{-3}{\sqrt{5}}, \frac{-6}{\sqrt{5}} \rangle \), the process is straightforward:

• Subtract each corresponding component: For example, \( \frac{4}{\sqrt{5}} - \left(\frac{-3}{\sqrt{5}}\right) \) becomes \( \frac{4}{\sqrt{5}} + \frac{3}{\sqrt{5}} \)
• Similarly, \( \frac{2}{\sqrt{5}} - \left(\frac{-6}{\sqrt{5}}\right) \) becomes \( \frac{2}{\sqrt{5}} + \frac{6}{\sqrt{5}} \)

The final result is the vector \( \langle \frac{7}{\sqrt{5}}, \frac{8}{\sqrt{5}} \rangle \). Subtraction provides the difference in position between the two vectors, which can be visualized as moving backward along the vector being subtracted.