Let a, b and c be distinct non-negative numbers. If the vectors a\widehat i + a\widehat j + c\widehat k,\,\,\widehat i + \widehat k and c\widehat i + c\widehat j + b\widehat k lie in a plane, then c is :

Vector Algebra PYQ — JEE Main 2005 · Shift 0 · Q91 with Solution

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Correct answer: A

$\vec v_1=(a,a,c),\qquad \vec v_2=(1,0,1),\qquad \vec v_3=(c,c,b).$

For three vectors to lie in the same plane, they must be coplanar. Hence their scalar triple product must be zero:

a & a & c\\ 1 & 0 & 1\\ c & c & b \end{vmatrix}=0.$$ 2. Expand the determinant: $$\begin{aligned} \Delta&=a\begin{vmatrix}0&1\\ c&b\end{vmatrix}-a\begin{vmatrix}1&1\\ c&b\end{vmatrix}+c\begin{vmatrix}1&0\\ c&c\end{vmatrix}\\[4pt] &=a(0\cdot b-1\cdot c)-a(1\cdot b-1\cdot c)+c(1\cdot c-0\cdot c)\\[4pt] &=-ac-a(b-c)+c^2. \end{aligned}$$ Now simplify: $$-ac-ab+ac+c^2= -ab+c^2.$$ Since the vectors are coplanar, $$-ab+c^2=0 \quad\Rightarrow\quad c^2=ab.$$ 3. Given that $a,b,c$ are non-negative numbers, $$c=\sqrt{ab}.$$ Thus $c$ is the geometric mean of $a$ and $b$. 4. Check options: - **A:** Geometric mean $=\sqrt{ab}$ ✔️ - **B:** Arithmetic mean $=\dfrac{a+b}{2}$ ✖️ - **C:** Equal to zero ✖️ (not necessarily) - **D:** Harmonic mean $=\dfrac{2ab}{a+b}$ ✖️ Therefore, the correct option is **A**.

View written solutionFree

Correct answer: A

$\vec v_1=(a,a,c),\qquad \vec v_2=(1,0,1),\qquad \vec v_3=(c,c,b).$

For three vectors to lie in the same plane, they must be coplanar. Hence their scalar triple product must be zero:

a & a & c\\ 1 & 0 & 1\\ c & c & b \end{vmatrix}=0.$$ 2. Expand the determinant: $$\begin{aligned} \Delta&=a\begin{vmatrix}0&1\\ c&b\end{vmatrix}-a\begin{vmatrix}1&1\\ c&b\end{vmatrix}+c\begin{vmatrix}1&0\\ c&c\end{vmatrix}\\[4pt] &=a(0\cdot b-1\cdot c)-a(1\cdot b-1\cdot c)+c(1\cdot c-0\cdot c)\\[4pt] &=-ac-a(b-c)+c^2. \end{aligned}$$ Now simplify: $$-ac-ab+ac+c^2= -ab+c^2.$$ Since the vectors are coplanar, $$-ab+c^2=0 \quad\Rightarrow\quad c^2=ab.$$ 3. Given that $a,b,c$ are non-negative numbers, $$c=\sqrt{ab}.$$ Thus $c$ is the geometric mean of $a$ and $b$. 4. Check options: - **A:** Geometric mean $=\sqrt{ab}$ ✔️ - **B:** Arithmetic mean $=\dfrac{a+b}{2}$ ✖️ - **C:** Equal to zero ✖️ (not necessarily) - **D:** Harmonic mean $=\dfrac{2ab}{a+b}$ ✖️ Therefore, the correct option is **A**.