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: Condition
The angle between a line and a plane is zero when the line makes $90^\circ$ with the normal, so $\cos 90^\circ = 0$.

Step 2: Compute
$\dfrac{3\cdot5 - 2b + 7\cdot3}{|\ldots|}$ is zero only if the numerator $36 - 2b = 0$.

Step 3: Result
$b = 18$.

Step 4: Why not others
$b = 13$ gives $36 - 26 = 10$, $b = 15$ gives 6 and $b = 16$ gives 4, none of which is zero.

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