If ⃗a ′=^i+^j,⃗b ′=^i+^j+2^k and ⃗c

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Question

If

⃗ a^{'} =^i +^j, ⃗ b^{'} =^i +^j + 2^k

and

⃗ c^{'} = 2^i +^j -^k

. Then altitude of the parallelopiped formed by the vectors

⃗ a, ⃗ b, ⃗ c

having base formed by

⃗ b

and

⃗ c

is

(⃗ a, ⃗ b, ⃗ c

and

{⃗ a}^{'}, {⃗ b}^{'}, {⃗ c}^{'}

are reciprocal system of vectors)

Options:

A . 1

B . 3√22

C . 1√6

D . 1√2

Answer: Option D
:
D
Volume of the parallelepiped formed by

⃗ a^{'}, ⃗ b^{'}, ⃗ c^{'}

is 4

∴

Volume of the parallelepiped formed by

⃗ a, ⃗ b, ⃗ c

is

\frac{1}{4}

⃗ b \times ⃗ c = \frac{1}{4} ⃗ a^{'} ∴ ∣ ∣ ⃗ b \times ⃗ c ∣ ∣ = \frac{\sqrt{2}}{4} = \frac{1}{2 \sqrt{2}}

∴

length of altitude =

\frac{1}{4} \times 2 \sqrt{2} = \frac{1}{\sqrt{2}}

.

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