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Applications of Dot Product

If r⃗ is an...

Question

If $\to r$ is any vectors then $(\to r \times^i) \times^i + (\to r \times^j) \times^j + (\to r \times^k) \times^k =$

A

\to r

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B

- 2 \to r

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C

4 \to r

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D

3 \to r

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Solution

The correct option is B $- 2 \to r$
$= (¯ ¯ ¯ r - ¯ i) ¯ i - ¯ ¯ ¯ r + (¯ ¯ ¯ r - ¯ j) ¯ j - ¯ ¯ ¯ r + (¯ ¯ ¯ r - ¯ ¯ ¯ k) ¯ ¯ ¯ k - ¯ ¯ ¯ r$
$= (r_{1}) ¯ i + (r_{2}) ¯ j + (r_{3}) ¯ ¯ ¯ k - 3 ¯ ¯ ¯ r$
$= + ¯ ¯ ¯ r - 3 ¯ ¯ ¯ r$
$= - 2 ¯ ¯ ¯ r$

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Similar questions

[\to p + 2 \to q + 3 \to r \to q + 2 \to r + 3 \to p \to r + 2 \to p + 3 \to q] = 54

\to p

\to q

\to r

are three vector then the Value of

∣ ∣ ∣ ∣ \begin{matrix} \to p . \to p & \to p . \to q & \to p . \to r \to p . \to q & \to q . \to q & \to q . \to r \to p . \to r & \to r . \to q & \to r . \to r \end{matrix} ∣ ∣ ∣ ∣

\to a = ˆ i + ˆ j

\to b = 2 ˆ i - ˆ k

\to r

satisfying the equations

\to r \times \to a = \to b \times \to a

\to r \times \to b = \to a \times \to b

\to r \times \to b = \to c \times \to b & \to r, \to a = 0

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

\to r

\to a, \to b, \to c

are three non-coplanar non-zero vectors and

\to r

is any vector, then

(\to a \times \to b) \times (\to r \times \to c) + (\to b \times \to c) \times (\to r \times \to a) + (\to c \times \to a) \times (\to r \times \to b) =

\to a, \to b, \to c

be three non-coplanar vectors and

\to r

be any arbitrary vector, then

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

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