Question

If the point P(x, 1, 4) lies on the line \(\overrightarrow{r}=\hat{i}+3\hat{j}+4\hat{k}+\lambda(2\hat{i}-\hat{j})\), then the value of x is equal to

• A. 2
• B. -2
• C. 3
• D. -3
• E. 5

Answer 1

Answer: (E)

Step 1: Parametric Equations of the Line  
The given line equation is: \[ \overrightarrow{r} = \hat{i} + 3\hat{j} + 4\hat{k} + \lambda(2\hat{i} - \hat{j}) \] This can be rewritten as: \[ x = 1 + 2\lambda, \quad y = 3 - \lambda, \quad z = 4 \]

Step 2: Use the Given Point P(x,1,4) 
Since the point \( P(x,1,4) \) lies on the line, we substitute \( y = 1 \) into the equation: \[ 1 = 3 - \lambda \]

Step 3: Solve for \( \lambda \) 
\[ \lambda = 3 - 1 = 2 \]

Step 4: Find \( x \) 
Substituting \( \lambda = 2 \) into the equation for \( x \): \[ x = 1 + 2(2) = 1 + 4 = 5 \]

Final Answer: \( 5 \).