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Applications of Cross Product

Solve the vec...

Question

Solve the vector equation $\to r \times \to b = \to a \times \to b, \to r . \to c = 0$ provided that $\to c$ is not perpendicular to $\to b$

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Solution

$\begin{matrix} G i v e n, ¯ ¯ ¯ r \times ¯ ¯ b = ¯ ¯ ¯ a \times ¯ ¯ b ¯ ¯ ¯ r . ¯ ¯ c = 0 ¯ ¯ c . ¯ ¯ b \neq 0 ¯ ¯ ¯ r \times ¯ ¯ b = ¯ ¯ ¯ a \times ¯ ¯ b t a k e n c r o s s i n c h ¯ ¯ c ¯ ¯ c \times (¯ ¯ ¯ r \times ¯ ¯ b) = ¯ ¯ c (¯ ¯ ¯ a \times ¯ ¯ b) (¯ ¯ c . ¯ ¯ b) ¯ ¯ ¯ r - (¯ ¯ c . ¯ ¯ ¯ r) ¯ ¯ b = (¯ ¯ c . ¯ ¯ b) ¯ ¯ ¯ a - (¯ ¯ ¯ a . ¯ ¯ c) ¯ ¯ b (¯ ¯ c . ¯ ¯ b) ¯ ¯ ¯ r - 0 = (¯ ¯ c . ¯ ¯ b) ¯ ¯ ¯ a - (¯ ¯ ¯ a . ¯ ¯ c) ¯ ¯ b ¯ ¯ ¯ r = ¯ ¯ ¯ a - \frac{(¯ ¯ ¯ a . ¯ c) ¯ b}{(¯ b . ¯ c)} \end{matrix}$

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Similar questions

\to r \cdot \to a = \to r \cdot \to b = \to r \cdot \to c = 0

\to a, \to b

\to c

are non-coplanar, then

\to a, \to b, \to c

be vectors of length

3, 4, 5

respectively. Let

\to a

be perpendicular to

\to b + \to c, \to b

\to c + \to a

\to c

\to a + \to b .

∣ ∣ \to a + \to b + \to c ∣ ∣

\to a, \to b, \to c

are three non-zero vectors such that

\to b

is not perpendicular to both

\to a

\to c

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

\to a

\to b

\to c

are three vectors and

\to a + \to b + \to c = 0

\to a \times \to b = \to b \times \to c = \to c \times \to a

Q. For any three vectors

\to a, \to b

\to c

[\to a + \to b, \to b + \to c, \to c + \to a] = 2 [\to a \to b \to c]

. Hence prove that the vectors

\to a + \to b, \to b + \to c, \to c + \to a

are coplanar. If and only if

\to a, \to b, \to c

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