Let a-b- and c- be three non-zero and non-coplanar vectors and p-q- and r- be three vectors given by p-a-b-2c- q-3a-2b-c- and r-a-4b-2c- If the volume of the parallelopiped determined by a-b- and c- is V1 and the volume of the parallelopiped determined by p-q- and r- is V2 then V2-V1-

The correct option is D 15:1 Volume of parallelopiped is given by V_1 = [a #160;b #160;c] V_2 = [p #160;qr] ∴ V_2 = (a + #160;b 2c ...

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Test for Collinearity of Vectors

Let a⃗,b⃗ a...

Question

Let $\to a, \to b$ and $\to c$ be three non-zero and non-coplanar vectors and $\to p, \to q$ and $\to r$ be three vectors given by $\to p = \to a + \to b - 2 \to c, \to q = 3 \to a - 2 \to b + \to c$ and $\to r = \to a - 4 \to b + 2 \to c$ . If the volume of the parallelopiped determined by $\to a, \to b$ and $\to c$ is $V_{1}$ and the volume of the parallelopiped determined by $\to p, \to q$ and $\to r$ is $V_{2}$ then $V_{2} : V_{1} =$

3 : 1

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7 : 1

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11 : 1

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15 : 1

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\to a, \to b, \to c

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}

\to q = \frac{\to c \times \to a}{[\to a \to b \to c]}

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

(2 \to a + 3 \to b + 4 \to c) . \to p + (2 \to b + 3 \to c + 4 \to a) . \to q +

(2 \to c + 3 \to a + 4 \to b) . \to r =

\to a, \to b, \to c

\to p, \to q, \to r

\to p = \frac{\to b \times \to c}{[\begin{matrix} \to b & \to c & \to a \end{matrix}]}

\to q = \frac{\to c \times \to a}{[\begin{matrix} \to c & \to a & \to b \end{matrix}]}

\to r = \frac{\to a \times \to b}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}

(\to a + \to b) . \to p + (\to b + \to c) . \to q + (\to c + \to a) . \to r

\to a, \to b, \to c

\to p, \to q, \to r

(\to a \times \to p) + (\to b \times \to q) + (\to c \times \to r)

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

¯ ¯ ¯ p, ¯ ¯ ¯ q, ¯ ¯ ¯ r

\to p = \frac{\to b \times \to c}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}, \to q = \frac{\to c \times \to a}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}, \to r = \frac{\to a \times \to b}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}

(\to a + \to b + \to c) . (\to p + \to q + \to r)

\to a, \to b

\to c

\to p, \to q

\to r

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

(\to a + \to b) \cdot \to p + (\to b + \to c) \cdot \to q + (\to c + \to a) \cdot \to r =

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