Question

If the angle between the lines $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{5-x}{-2}=\frac{7y-14}{P}=\frac{z-3}{4}$ is $\cos^{-1} \frac{2}{3}$ the 'P' is equal to _____.

Answer 1

$\frac{7}{2}$

Answer 2

1. Identify direction ratios of the first line as $(2, 2, 1)$.
2. Rewrite the second line as $\frac{x-5}{2} = \frac{y-2}{P/7} = \frac{z-3}{4}$ and identify its direction ratios as $(2, P/7, 4)$.
3. Use the formula for the angle $\theta$ between two lines: $\cos \theta = \frac{|\vec{d_1} \cdot \vec{d_2}|}{|\vec{d_1}| |\vec{d_2}|}$.
4. Substitute the given $\cos \theta = 2/3$ and the direction ratios into the formula.
5. Calculate the dot product: $2(2) + 2(P/7) + 1(4) = 8 + 2P/7$.
6. Calculate the magnitudes: $|\vec{d_1}| = \sqrt{2^2+2^2+1^2} = 3$ and $|\vec{d_2}| = \sqrt{2^2+(P/7)^2+4^2} = \sqrt{20+P^2/49}$.
7. Set up the equation: $\frac{2}{3} = \frac{|8+2P/7|}{3 \sqrt{20+P^2/49}}$.
8. Simplify and square both sides to solve for $P$: $4 = \frac{(8+2P/7)^2}{20+P^2/49}$.
9. This leads to $4(980+P^2) = (56+2P)^2$, which simplifies to $224P = 784$.
10. Solve for $P = 784/224 = 7/2$.