Answer :
To determine the angle between the rays [tex]\( R_p \)[/tex] and [tex]\( R_q \)[/tex], we can follow these steps:
1. Identify the Direction Vectors:
We have two points:
- Point [tex]\( p = \left( \frac{3}{13}, \frac{\sqrt{160}}{13} \right) \)[/tex]
- Point [tex]\( q = \left( \frac{18}{20}, \frac{\sqrt{76}}{20} \right) \)[/tex]
2. Create Vectors:
The direction vectors for rays [tex]\( R_p \)[/tex] and [tex]\( R_q \)[/tex] originating from the origin (0,0) passing through points [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
- Vector [tex]\( \mathbf{p} = \left( \frac{3}{13}, \frac{\sqrt{160}}{13} \right) \)[/tex]
- Vector [tex]\( \mathbf{q} = \left( \frac{18}{20}, \frac{\sqrt{76}}{20} \right) \)[/tex]
3. Calculate the Dot Product:
The dot product ([tex]\(\cdot\)[/tex]) of two vectors [tex]\( \mathbf{a} = (a_1, a_2) \)[/tex] and [tex]\( \mathbf{b} = (b_1, b_2) \)[/tex] is given by:
[tex]\[
\mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2
\][/tex]
For our vectors:
[tex]\[
\mathbf{p} \cdot \mathbf{q} \approx 0.6318
\][/tex]
4. Calculate Magnitudes of the Vectors:
The magnitude ([tex]\(||\mathbf{a}||\)[/tex]) of a vector [tex]\(\mathbf{a} = (a_1, a_2)\)[/tex] is:
[tex]\[
||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2}
\][/tex]
For both [tex]\(\mathbf{p}\)[/tex] and [tex]\(\mathbf{q}\)[/tex], the magnitudes are 1 since:
[tex]\[
||\mathbf{p}|| = ||\mathbf{q}|| = 1
\][/tex]
5. Calculate the Cosine of the Angle:
The cosine of the angle [tex]\(\theta\)[/tex] between two vectors is given by:
[tex]\[
\cos \theta = \frac{\mathbf{p} \cdot \mathbf{q}}{||\mathbf{p}|| \cdot ||\mathbf{q}||}
\][/tex]
Substituting the known values:
[tex]\[
\cos \theta \approx 0.6318
\][/tex]
6. Find the Angle in Radians and Degrees:
To find the angle (in radians), we use the arccos function:
[tex]\[
\theta \approx \arccos(0.6318) \approx 0.887 \text{ radians}
\][/tex]
To convert radians to degrees:
[tex]\[
\theta \approx 0.887 \times \frac{180}{\pi} \approx 50.82 \text{ degrees}
\][/tex]
Thus, the angle [tex]\(\angle R_p R_q\)[/tex] is approximately [tex]\(50.82^\circ\)[/tex].
1. Identify the Direction Vectors:
We have two points:
- Point [tex]\( p = \left( \frac{3}{13}, \frac{\sqrt{160}}{13} \right) \)[/tex]
- Point [tex]\( q = \left( \frac{18}{20}, \frac{\sqrt{76}}{20} \right) \)[/tex]
2. Create Vectors:
The direction vectors for rays [tex]\( R_p \)[/tex] and [tex]\( R_q \)[/tex] originating from the origin (0,0) passing through points [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
- Vector [tex]\( \mathbf{p} = \left( \frac{3}{13}, \frac{\sqrt{160}}{13} \right) \)[/tex]
- Vector [tex]\( \mathbf{q} = \left( \frac{18}{20}, \frac{\sqrt{76}}{20} \right) \)[/tex]
3. Calculate the Dot Product:
The dot product ([tex]\(\cdot\)[/tex]) of two vectors [tex]\( \mathbf{a} = (a_1, a_2) \)[/tex] and [tex]\( \mathbf{b} = (b_1, b_2) \)[/tex] is given by:
[tex]\[
\mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2
\][/tex]
For our vectors:
[tex]\[
\mathbf{p} \cdot \mathbf{q} \approx 0.6318
\][/tex]
4. Calculate Magnitudes of the Vectors:
The magnitude ([tex]\(||\mathbf{a}||\)[/tex]) of a vector [tex]\(\mathbf{a} = (a_1, a_2)\)[/tex] is:
[tex]\[
||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2}
\][/tex]
For both [tex]\(\mathbf{p}\)[/tex] and [tex]\(\mathbf{q}\)[/tex], the magnitudes are 1 since:
[tex]\[
||\mathbf{p}|| = ||\mathbf{q}|| = 1
\][/tex]
5. Calculate the Cosine of the Angle:
The cosine of the angle [tex]\(\theta\)[/tex] between two vectors is given by:
[tex]\[
\cos \theta = \frac{\mathbf{p} \cdot \mathbf{q}}{||\mathbf{p}|| \cdot ||\mathbf{q}||}
\][/tex]
Substituting the known values:
[tex]\[
\cos \theta \approx 0.6318
\][/tex]
6. Find the Angle in Radians and Degrees:
To find the angle (in radians), we use the arccos function:
[tex]\[
\theta \approx \arccos(0.6318) \approx 0.887 \text{ radians}
\][/tex]
To convert radians to degrees:
[tex]\[
\theta \approx 0.887 \times \frac{180}{\pi} \approx 50.82 \text{ degrees}
\][/tex]
Thus, the angle [tex]\(\angle R_p R_q\)[/tex] is approximately [tex]\(50.82^\circ\)[/tex].
