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Vector Addition

If a⃗= 2î-3...

Question

If $\to a = 2^i - 3^j + 5^k, \to b = 3^i - 4^j + 5^k$ and $\to c = 5^i - 3^j - 2^k$ , then the volume of the parallelopiped with conterminous edges $\to a + \to b, \to b + \to c, \to c + \to a$ is

A

20

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B

10

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C

16

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D

0

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Solution

The correct option is B $16$
$¯ ¯ ¯ a + ¯ ¯ b = 5^i - 7^j + 10^k$
$¯ ¯ b + ¯ ¯ c = 8^i - 7^j + 3^k$
$¯ ¯ c + ¯ ¯ ¯ a = 7^i - 6^j + 3^k$
Volume of parallelopipe = $[¯ ¯ ¯ a + ¯ ¯ b ¯ ¯ b + ¯ ¯ c ¯ ¯ c + ¯ ¯ ¯ a]$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 5 & - 7 & 10 8 & - 7 & 3 7 & - 6 & 3 \end{matrix} ∣ ∣ ∣ ∣$
$\Rightarrow 5 [(- 21 + 18)] + 7 (24 - 21) + 10 (- 48 + 49)$
$\Rightarrow - 15 + 21 + 10$
$\Rightarrow 16$ cubic units

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\to a = 3^i - 2^j +^k,

\to b = 2^i - 4^j - 3^k, \to c = -^i + 2^j + 2^k

\to a + \to b + \to c = 4^i - 4^j

\to a =^i -^j + 2^k, \to b = 2^i + 3^j +^k, \to c =^i -^k

, then magnitude of

\to a + 2 \to b - 3 \to c

\sqrt{78}

\to a = 2^i + 4^j - 5^k, \to b =^i +^j +^k, \to c =^j + 2^k

, then the unit vectors parallel to

\to a + \to b + \to c

\to a = 3^i + 2^j + 2^k, \to b = -^i + 3^j -^k, \to c =^i +^j +^k

, then find the value of

\to a \times (\to b \times \to c)

(\to a \times \to b) \times \to c

\to a = 3^i - 2^j +^k, \to b = 2^i - 4^j - 3^k

\to c = - 1^i + 2^j + 2^k

\to a + \to b + \to c =

\to a = 2^i - 5^j + 8^k, \to b =^i - 3^j -^k, \to c = - 3^i - 2^j -^k

| \to a + \to b + \to c |

.

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