Question

Let \(\overset{⃗}{r}\cdot (3\hat{i}-2\hat{j}+7\hat{k}) = 32\) is the equation of a plane and the line having direction ratios \((5,b,3)\) is parallel to the plane, then the value of b is...

• A. \(b = 13\)
• B. \(b = 15\)
• C. \(b = 16\)
• D. \(b = 18\)

Hint:

A line is parallel to a plane when its direction is perpendicular to the plane normal.

Answer 1

The Correct Option is D

Step 1: Understand the concept
A line with direction ratios \((l, m, n)\) is parallel to the plane \(\vec{r}\cdot\vec{N} = d\) when \(\vec{N}\) is perpendicular to the direction, that is, \(\vec{N}\cdot(l, m, n) = 0\).

Step 2: Identify the vectors
The normal is \(\vec{N} = (3, -2, 7)\) and the direction is \((5, b, 3)\).

Step 3: Set the dot product to zero
\[ 3(5) + (-2)(b) + 7(3) = 0 \Rightarrow 15 - 2b + 21 = 0 \]

Step 4: Solve
\(2b = 36\), so \(b = 18\), option (D). Check: \(15 - 36 + 21 = 0\).

Final Answer:
The value of b is 18. This is option (D). \[ \boxed{\text{(D) }b=18} \]