If the line $\frac{x+4}{a}=\frac{y-2}{b}=\frac{z+1}{c}$ passes through the point $(-3,1,-2)$ and its direction is chosen such that it makes an acute angle $\gamma$ with the positive direction of Z- axis, then value of $\cos \gamma$ is:

Question

The angle between the lines whose direction cosines satisfy the equations: \[ l + m + n = 0 \quad \text{and} \quad m^2 + n^2 - l^2 = 0 \] Find the angle between the two lines.

Answer 1

Step 1: Idea
Two points on the line give its direction: $(-4,2,-1)$ and $(-3,1,-2)$.

Step 2: Direction Vector
$(-3,1,-2)-(-4,2,-1)=(1,-1,-1)$.

Step 3: Cosine
The z-component of the unit vector is $\mp\dfrac1{\sqrt3}$. Choosing the direction $(-1,1,1)$ makes the angle with the positive Z-axis acute, so $\cos\gamma=\dfrac1{\sqrt3}$.

Final Answer:
Taking the direction with a positive z-component gives 1 over root 3. \[ \boxed{\text{(C) }\dfrac{1}{\sqrt3}} \]

If the line \(\frac{x+4}{a}=\frac{y-2}{b}=\frac{z+1}{c}\) passes through the point \((-3,1,-2)\) and its direction is chosen such that it makes an acute angle \(\gamma\) with the positive direction of Z- axis, then value of \(\cos \gamma\) is:

Show Hint

The Correct Option is C

Solution and Explanation

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