If the angle between the lines, `x/2=y/2=z/1 and (5-x)/(-2) = (7y-14)/(p ) = (z-3)/(4) is cos ^(-1) ((2)/(3)),` then P is equal to

Options

A. `-4/7`
B. `7/2`
C. `-7/4`
D. `7/2`

Answer

To solve the problem, we need to find the value of \( p \) given the angle between two lines. Let's break it down step by step.

### Step 1: Identify the direction vectors of the lines

The first line is given in symmetric form:
\[ \frac{x}{2} = \frac{y}{2} = \frac{z}{1} \]
From this, we can identify the direction vector \( \vec{d_1} \) as:
\[ \vec{d_1} = (2, 2, 1) \]

The second line is given as:
\[ \frac{5 - x}{-2} = \frac{7y - 14}{p} = \frac{z - 3}{4} \]
We can rewrite this in standard form:
\[ 5 - x = -2t \implies x = 5 + 2t \]
\[ 7y - 14 = pt \implies y = 2 + \frac{pt}{7} \]
\[ z - 3 = 4t \implies z = 3 + 4t \]
Thus, the direction vector \( \vec{d_2} \) becomes:
\[ \vec{d_2} = (2, \frac{p}{7}, 4) \]

### Step 2: Use the formula for the angle between two lines

The angle \( \theta \) between two lines can be found using the dot product of their direction vectors:
\[ \cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|} \]
We know from the problem that:
\[ \cos \theta = \frac{2}{3} \]

### Step 3: Calculate the dot product \( \vec{d_1} \cdot \vec{d_2} \)

Calculating the dot product:
\[ \vec{d_1} \cdot \vec{d_2} = (2)(2) + (2)\left(\frac{p}{7}\right) + (1)(4) = 4 + \frac{2p}{7} + 4 = 8 + \frac{2p}{7} \]

### Step 4: Calculate the magnitudes of the direction vectors

Calculating the magnitude of \( \vec{d_1} \):
\[ |\vec{d_1}| = \sqrt{2^2 + 2^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

Calculating the magnitude of \( \vec{d_2} \):
\[ |\vec{d_2}| = \sqrt{2^2 + \left(\frac{p}{7}\right)^2 + 4^2} = \sqrt{4 + \frac{p^2}{49} + 16} = \sqrt{20 + \frac{p^2}{49}} \]

### Step 5: Set up the equation using the cosine formula

Substituting into the cosine formula:
\[ \frac{8 + \frac{2p}{7}}{3 \sqrt{20 + \frac{p^2}{49}}} = \frac{2}{3} \]

### Step 6: Cross-multiply and simplify

Cross-multiplying gives:
\[ 8 + \frac{2p}{7} = 2 \sqrt{20 + \frac{p^2}{49}} \]
Squaring both sides:
\[ \left(8 + \frac{2p}{7}\right)^2 = 4\left(20 + \frac{p^2}{49}\right) \]

### Step 7: Expand and simplify the equation

Expanding the left side:
\[ 64 + 2 \cdot 8 \cdot \frac{2p}{7} + \left(\frac{2p}{7}\right)^2 = 4 \cdot 20 + \frac{4p^2}{49} \]
\[ 64 + \frac{32p}{7} + \frac{4p^2}{49} = 80 + \frac{4p^2}{49} \]
The \( \frac{4p^2}{49} \) cancels out:
\[ 64 + \frac{32p}{7} = 80 \]
Rearranging gives:
\[ \frac{32p}{7} = 80 - 64 = 16 \]
Multiplying both sides by 7:
\[ 32p = 112 \]
Dividing by 32:
\[ p = \frac{112}{32} = 3.5 \]

### Final Answer
Thus, the value of \( p \) is:
\[ p = 7 \]