If the angle between the lines x2= y2= z1 and 5-x-2=7y-14P=z-34 is ^-123, then P is equal …

Question

If the angle between the lines $\dfrac x2=\dfrac y2=\dfrac z1$ and $\dfrac{5-x}{-2}=\dfrac{7y-14}P=\dfrac{z-3}4$ is $\cos^{-1}\dfrac23$, then $P$ is equal to

Accepted Answer

Second line direction: $\left(2,\dfrac P7,4\right)$. Setting $\cos\theta=\dfrac23$ with the first line's direction $(2,2,1)$:
$$\frac{4+\tfrac{2P}7+4}{3\sqrt{4+\tfrac{P^2}{49}+16}}=\frac23\implies 4+\frac P7=\sqrt{20+\frac{P^2}{49}}.$$
Squaring: $16+\dfrac{8P}7+\dfrac{P^2}{49}=20+\dfrac{P^2}{49}\implies \dfrac{8P}7=4\implies P=\dfrac72.$

Solutions · 1