Problem 40 - Precalculus with Limits

Question

In Exercises 37-46, find the angle \(\theta\) (in radians and degrees)between the lines. \(2x - y = 2\) \(4x + 3y = 24\)

Step-by-Step Solution

Answer

\(\theta\) (in radians) = \(arccos(\frac{11\sqrt{5}}{25})\) and in degrees this would be \(\theta = arccos(\frac{11\sqrt{5}}{25}) \cdot \frac {180}{\pi}\).

Step 1: Find The Direction Vectors

The direction vectors for the lines can be determined using the ratio \(a_x/a_y = -A/B\). For the first line, \(a1 = (-1, 2)\) and for the second line, \(a2 = (-3, 4)\).

Step 2: Calculate The Dot Product Of The Vectors

The dot product of \(a1\) and \(a2\) is calculated as follows: \(a1 \cdot a2 = a_{1x} \cdot a_{2x} + a_{1y} \cdot a_{2y} = -1 \cdot -3 + 2 \cdot 4 = 11\).

Step 3: Calculate The Norms Of The Vectors

The norm (or length) of a vector \(a\) is calculated as follows: \(||a|| = \sqrt{a_x^2 + a_y^2}\). So for \(a1\), it will be \(||a1|| = \sqrt{(-1)^2 + 2^2} = \sqrt{5}\) and for \(a2\), it will be \(||a2|| = \sqrt{(-3)^2 + 4^2} = \sqrt{25} = 5\).

Step 4: Calculate The Cosine Of The Angle

The cosine of the angle \(\theta\) between \(a1\) and \(a2\) is \(cos \theta = \frac {a1 \cdot a2} {||a1|| ||a2||} = \frac{11}{5 \cdot \sqrt{5}} = \frac{11\sqrt{5}}{25}\).

Step 5: Calculate The Angle In Radians And Convert To Degrees

Finally, use the inverse cosine function to find \(\theta\) in radians: \(\theta = arccos(\frac{11\sqrt{5}}{25})\). Convert this into degrees by multiplying by \(\frac {180}{\pi}\).

Key Concepts

Direction Vectors

In geometry, understanding the direction of a line is crucial. The direction vector of a line tells us where the line points. If you have a straight line equation like this: \( Ax + By = C \), a direction vector can be expressed as \( \mathbf{a} = (a_x, a_y) \). This vector will be perpendicular to the line's normal vector, \( (-B, A) \). In the example given, the direction vectors are derived from the line equations as \( \mathbf{a1} = (-1, 2) \) and \( \mathbf{a2} = (-3, 4) \).

• Direction Vector for Line 1: From the equation \( 2x - y = 2 \), rearrange to get \(-1\) as the x-component and \(2\) as the y-component, resulting in the vector \( (-1, 2) \).
• Direction Vector for Line 2: Similarly, from \( 4x + 3y = 24 \), we obtain \(-3\) and \(4\) for the components, forming the vector \( (-3, 4) \).

By using these vectors, we can investigate the relationships between lines, including angles and intersections.

Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar. It's useful for understanding the angle between vectors. For two vectors \( \mathbf{a} = (a_x, a_y) \) and \( \mathbf{b} = (b_x, b_y) \), the dot product is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = a_x \cdot b_x + a_y \cdot b_y \].In our problem, we find the dot product of the vectors \( \mathbf{a1} = (-1, 2) \) and \( \mathbf{a2} = (-3, 4) \):

• Calculate each component: \((-1) \cdot (-3) = 3\) and \(2 \cdot 4 = 8\).
• Add these results: \(3 + 8 = 11\).

The dot product tells us the angle’s cosine value when divided by the product of the vectors' norms, but more on that later. Here, it tells us \( \mathbf{a1} \cdot \mathbf{a2} = 11 \). This is the first step in finding the angle between the two lines.

Norm of a Vector

The norm of a vector, often simply called the "length" or "magnitude," tells us how long the vector is. For a vector \( \mathbf{a} = (a_x, a_y) \), its norm is calculated using the formula:\[ ||\mathbf{a}|| = \sqrt{a_x^2 + a_y^2} \].Calculating the norms for our direction vectors:

• For \( \mathbf{a1} = (-1, 2) \): \( ||\mathbf{a1}|| = \sqrt{(-1)^2 + 2^2} = \sqrt{5} \).
• For \( \mathbf{a2} = (-3, 4) \): \( ||\mathbf{a2}|| = \sqrt{(-3)^2 + 4^2} = \sqrt{25} = 5 \).

This measurement is key when calculating the cosine of the angle between two vectors. By knowing the norms, we can assess how these vectors scale, which is essential in proceeding with finding the angles.

Inverse Cosine Function

The inverse cosine function, denoted as \( \arccos \), is used to determine the angle from the cosine value. In our context, once we have the dot product and the norms, we can find the cosine of the angle between vectors. The formula is:\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \, ||\mathbf{b}||} \].With the given values:

• Dot Product: \(11\)
• Norm of \( \mathbf{a1} \): \( \sqrt{5} \)
• Norm of \( \mathbf{a2} \): \(5\)
• Cosine of \( \theta \): \(\frac{11}{5 \cdot \sqrt{5}} = \frac{11\sqrt{5}}{25}\)

To find the angle \( \theta \), apply the inverse cosine function:\[ \theta = \arccos\left(\frac{11\sqrt{5}}{25}\right) \].Convert from radians to degrees by multiplying with \( \frac{180}{\pi} \). The inverse cosine lets us bridge from dimensional analysis to actual angular measurements, providing a tangible sense of the orientation between our vectors.