Question

If \(\vec{r}=4\hat{i}+2\hat{j}+7\hat{k}+\lambda (3\hat{i}-5\hat{k})\) is the vector equation of the line passing through \((7,2,2)\), where \(\lambda\) is parameter, then the value of \(\lambda\) is...

• A. \(-1\)
• B. \(2\)
• C. \(-2\)
• D. \(1\)

Hint:

Equate the position vector of the given point with the line equation and compare components.

Answer 1

The Correct Option is D

Step 1: Parametric Form
The line is $x=4+3\lambda$, $y=2$, $z=7-5\lambda$.

Step 2: Substitute the Point
Put $x=7$ to get $3\lambda=3$, and $z=2$ to get $5\lambda=5$.

Step 3: Conclude
Both give $\lambda=1$, and $y=2$ already matches.

Final Answer:
The parameter value is 1. \[ \boxed{\text{(D) }1} \]