Question

If \(\overset{̄}{a}\) and \(\overset{̄}{b}\) are unit vectors perpendicular to each other, then \([\overset{̄}{a}+(\overset{̄}{a}\times \overset{̄}{b})\,\overset{̄}{b}+(\overset{̄}{a}\times \overset{̄}{b})\,(\overset{̄}{a}\times \overset{̄}{b})] = \cdots\)

• A. \(-1\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

Subtract the third vector from the first two rows; the triple product is unchanged.

Answer 1

The Correct Option is B

Step 1: Choose a concrete example:
Take $\bar a = \hat i$, $\bar b = \hat j$. Then $\bar c = \bar a\times\bar b = \hat k$.

Step 2: Write the vectors:
$\bar a + \bar c = (1,0,1)$, $\bar b + \bar c = (0,1,1)$, $\bar c = (0,0,1)$.

Step 3: Determinant:
$\begin{vmatrix}1&0&1\\0&1&1\\0&0&1\end{vmatrix}$ is an upper triangular determinant equal to $1\cdot1\cdot1 = 1$.

Final Answer:
The value is 1, option (B). \[ \boxed{1} \]