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Test for Collinearity of Vectors

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Question

From the following system of unknown vectors $(\to a$ and $\to b$ are given vectors $) \to x + \to y = \to a, \to x \times \to y = \to b, \to x . \to a = 1$ and
$\to x = \frac{\to a + λ (\to a \times \to b)}{| \to a |^{2}}$ . Then $λ$ is

A

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B

- 1

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C

2

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D

- 2

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Solution

The correct option is A $1$
$\to x + \to y = \to a \dots (i) \to x \times \to y = \to b \dots (i i) \to x . \to a = 1 \dots (i i i)$
$| \to a |^{2} \to x = \to a + λ (\to a \times \to b) \dots (i v)$
Multiplying equation $(i)$ by dot product of $\to a$
$\to x . \to a + \to y . \to a = \to a . \to a$
$1 + \to y . \to a = | \to a |^{2} \dots (v)$
Now, taking cross product of equation $(i i)$ with $\to a$
$\to a \times (\to x \times \to y) = \to a \times \to b$
$(\to a . \to y) \to x - (\to a . \to x) \to y = \to a \times \to b$
$(\to a . \to y) \to x - (1) \to y = \to a \times \to b$ From $(i i i)$
$(\to a . \to y) \to x - (\to a - \to x) = \to a \times \to b$ From $(i)$
$(\to a . \to y) \to x + \to x = \to a + \to a \times \to b$
$((\to a . \to y) + 1) \to x = \to a + \to a \times \to b$
putting from equation $(v)$
$| \to a |^{2} \to x = \to a + \to a \times \to b \dots (v i)$
Comparing equation $(i v), (v i)$
$λ = 1$

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Similar questions

Q. Let two vectors

\to x

\to y

satisfy the following conditions :

\to x + \to y = \to a; \to x \times \to y = \to b \times \to a, \to x . \to a = 1

\to a

\to b

are non-zero perpendicular vectors, then

\to x, \to y

\to z

be unit vectors such that

\to x + \to y + \to z = \to a, \to x \times (\to y \times \to z) = \to b, (\to x \times \to y) \times \to z = \to c

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

| \to a | = 2

.

Then which of the following option(s) is/are CORRECT ?

\to x, \to y, \to z

are units vectors such that

\to x + \to y + \to z = \to a

(\to x \times \to y) \times \to z = \to b

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

| \to a | = 2,

then which of the following is FALSE?

\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1

\to y \cdot \to z = 1

\to x

\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1

\to y \cdot \to z = 1

\to y

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Test for Collinearity of Vectors

Standard XII Mathematics

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