Question

The value of \(b\) such that the scalar product of the vector \(\hat{i}+\hat{j}+\hat{k}\) with the unit vector parallel to the sum of the vectors \(2\hat{i}+4\hat{j}-5\hat{k}\) and \(b\hat{i}+2\hat{j}+3\hat{k}\) is one, is...

• A. \(-2\)
• B. \(-1\)
• C. \(0\)
• D. \(1\)

Hint:

Altitude equals the volume divided by the base area.

Answer 1

The Correct Option is D

Step 1: Project onto the normal:
The altitude is the component of $\vec c$ along the unit normal to the base, $\hat n = \frac{\vec a\times\vec b}{|\vec a\times\vec b|}$.

Step 2: Compute:
$\vec a\times\vec b = (-5, 3, 2)$, $|\cdot| = \sqrt{38}$. $h = |\vec c\cdot\hat n| = \frac{|-5 + 3 + 6|}{\sqrt{38}} = \frac{4}{\sqrt{38}} = \frac{2\sqrt2}{\sqrt{19}}$. Option (A).

Final Answer:
Option (A). \[ \boxed{\frac{2\sqrt{2}}{\sqrt{19}}} \]