Volume of parallelopiped formed by vectors $\to a \times \to b, \to b \times \to c$ and $\to c \times \to a$ is $36$ cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

$\begin{matrix} C o l u m n 1 & C o l u m n 2 a. Volume of parallelopiped formed by vectors & p. 0 cubic units \to a, \to b and \to c is b. Volume of tetrahedron formed by vectors & q. 12 cubic units \to a, \to b and \to c is c. Volume of parallelopiped formed by vectors & r. 6 cubic units \to a + \to b, \to b + \to c and \to c + \to a is d. Volume of parallelopiped formed by vectors & s. 1 cubic unit \to a - \to b, \to b - \to c and \to c - \to a is \end{matrix}$

a - r, b - s, c - q, d - p

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a - p, b - s, c - q, d - r

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a - r, b - q, c - s, d - p

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a - s, b - r, c - q, d - p

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Solution

The correct option is A $a - r, b - s, c - q, d - p$
$[\to a \times \to b \to b \times \to c \to c \times \to a] = 36$
$[\to a \to b \to c] = 6$
$\Rightarrow$ Volume of tetrahedron formed by vectors $\to a, \to b$ and $\to c$ is $\frac{1}{6} [\to a \to b \to c] = 1$
$[\to a + \to b \to b + \to c \to c + \to a] = 2 [\to a \to b \to c] = 12$
$\to a - \to b, \to b - \to c, \to c - \to a$ are coplanar
$[\to a - \to b \to b - \to c \to c - \to a] = 0$

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Similar questions

\to a \times \to b, \to b \times \to c

\to c \times \to a

36

\begin{matrix} C o l u m n 1 & C o l u m n 2 a. Volume of parallelopiped formed by vectors & p. 0 cubic units \to a, \to b and \to c is b. Volume of tetrahedron formed by vectors & q. 12 cubic units \to a, \to b and \to c is c. Volume of parallelopiped formed by vectors & r. 6 cubic units \to a + \to b, \to b + \to c and \to c + \to a is d. Volume of parallelopiped formed by vectors & s. 1 cubic unit \to a - \to b, \to b - \to c and \to c - \to a is \end{matrix}

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\begin{matrix} C o l u m n 1 & C o l u m n 2 a. The length of the common chord of two circles of radii 3 and & p. 1 4 units which intersect orthogonally is \frac{k}{5}, then k is equal to b. The circumference of the circle x^{2} + y^{2} + 4 x + 12 y + p = 0 & q. 24 is bisected by the circle x^{2} + y^{2} - 2 x + 8 y - q = 0, then p + q is equal to c. Number of distinct chords of the circle 2 x (x - \sqrt{2}) + y (2 y - 1) & r. 32 = 0 chords are passing through the point (\sqrt{2}, \frac{1}{2}) and are bisected on x-axis is d. One of the diameters of the circle circumscribing the rectangle & s. 36 A B C D is 4 y = x + 7 . If A and B are the points (- 3, 4) and (5, 4) respectively, then the area of rectangle is \end{matrix}

Column I

Column II

\begin{matrix} C o l u m n I & C o l u m n I I (T y p e o f △ A B C) a. cot \frac{A}{2} = \frac{b + c}{a} & p. always right angled b. a tan A + b tan B = (a + b) tan \frac{A + B}{2} & q. always isosceles c. a cos A = b cos B & r. may be right angled d. cos A = \frac{sin B}{2 sin C} & s. may be right angled isosceles \end{matrix}

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