Question

If \(\overset{⃗}{a},\overset{⃗}{b},\overset{⃗}{c}\) are three non-coplanar vectors and \(\overset{⃗}{p},\overset{⃗}{q},\overset{⃗}{r}\) are defined as \(\overset{⃗}{p} = \frac{\overset{⃗}{b}\times \overset{⃗}{c}}{[\overset{⃗}{a} \overset{⃗}{b} \overset{⃗}{c}]},\overset{⃗}{q} = \frac{\overset{⃗}{c}\times \overset{⃗}{a}}{[\overset{⃗}{a} \overset{⃗}{b} \overset{⃗}{c}]},\overset{⃗}{r} = \frac{\overset{⃗}{a}\times \overset{⃗}{b}}{[\overset{⃗}{a} \overset{⃗}{b} \overset{⃗}{c}]}\), then \([(\overset{⃗}{a}+\overset{⃗}{b})\cdot \overset{⃗}{p}+(\overset{⃗}{b}+\overset{⃗}{c})\cdot \overset{⃗}{q}+(\overset{⃗}{c}+\overset{⃗}{a})\cdot \overset{⃗}{r}]\) is equal to

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

The vectors p, q, r form the reciprocal system, so a dot p equals 1 and b dot p equals 0.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The vectors \(\vec p,\vec q,\vec r\) are the reciprocal system to \(\vec a,\vec b,\vec c\). They satisfy
\[ \vec a\cdot\vec p=\vec b\cdot\vec q=\vec c\cdot\vec r=1,\qquad \vec b\cdot\vec p=\vec c\cdot\vec p=\vec a\cdot\vec q=\vec c\cdot\vec q=\vec a\cdot\vec r=\vec b\cdot\vec r=0 \]
The zeros arise because \(\vec b\times\vec c\) is perpendicular to \(\vec b\) and \(\vec c\).

Step 2: Evaluate each term
\((\vec a+\vec b)\cdot\vec p=1+0=1\).
\((\vec b+\vec c)\cdot\vec q=1+0=1\).
\((\vec c+\vec a)\cdot\vec r=1+0=1\).

Step 3: Sum
\[ 1+1+1=3 \]
This is option (D).

Final Answer:
Each bracket equals 1 for the reciprocal system, so the sum is 3, option (D). \[ \boxed{3} \]