Question

Lines \(\overset{⃗}{r} = \overset{⃗}{a}+λ\overset{⃗}{b}\) and \(\overset{⃗}{r} = \overset{⃗}{b}+μ\overset{⃗}{a}\) intersect at point \((2,4,-4)\). If \(|\overset{⃗}{a}-\overset{⃗}{b}| = 4\), then \(\overset{⃗}{a}\cdot \overset{⃗}{b} =\)

• A. \(5\)
• B. \(10\)
• C. \(-5\)
• D. \(-10\)

Hint:

At the intersection a + lambda b equals b + mu a, which forces lambda and mu to be 1 for non-parallel vectors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
At the common point, \(\vec a+\lambda\vec b=\vec b+\mu\vec a\). Rearranged, \((1-\mu)\vec a=(1-\lambda)\vec b\).

Step 2: Independence
The vectors \(\vec a\) and \(\vec b\) are not parallel (otherwise the lines would not be distinct lines in general). So both coefficients must vanish: \(\mu=1\) and \(\lambda=1\).

Step 3: Point of intersection
\[ \vec a+\vec b=(2,4,-4)\Rightarrow|\vec a+\vec b|^2=4+16+16=36 \]

Step 4: Use the given length
\[ |\vec a-\vec b|^2=16 \]

Step 5: Subtract
\[ |\vec a+\vec b|^2-|\vec a-\vec b|^2=4\,\vec a\cdot\vec b \]
\[ 36-16=20=4\,\vec a\cdot\vec b\Rightarrow\vec a\cdot\vec b=5 \]
This is option (A).

Final Answer:
The intersection point is a + b, so the squares of the sum and difference give a dot b = 5, option (A). \[ \boxed{5} \]