Linear Depend...

Linear Dependence and Independence of Vectors

Trending Questions

If the vectors $\to a$ and $\to b$ are perpendicular to each other, then a vector $\to v$ in terms of $\to a$ and $\to b$ satisfying the equations
$\to v . \to a = 0, \to v . \to b = 1 a n d [\to v \to a \to b] = 1$ is

$\frac{\to b}{| \to b |^{2}} + \frac{\to a \times \to b}{| \to a \times \to b |^{2}}$
$\frac{\to b}{| \to b |} + \frac{\to a \times \to b}{| \to a \times \to b |^{2}}$
$\frac{\to b}{| \to b |^{2}} + \frac{\to a \times \to b}{| \to a \times \to b |}$
None of these

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Q. Let

\to a, \to b

and

\to c

be three unit vectors, out of which vectors

\to b

and

\to c

are non-parallel. If

α

and

β

are the angles which vector

\to a

makes with vectors

\to b

and

\to c

respectively and

\to a \times (\to b \times \to c) = \frac{1}{2} \to b

, then

| α - β |

is equal to :

$60^{\circ}$
$45^{\circ}$
$30^{\circ}$
$90^{\circ}$

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Q. System of vectors

a_{1}, a_{2}, . . . . . . . a_{n}

is said to be linearly dependent if there exists a system of scalars

(c_{1}, c_{2} - - - c_{n} s u c h t h a t c_{1} ¯ a + c_{2}_{2} + . . . c_{n}_{n} = ¯ 0

True
False

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Q. If

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

and

\to c =^i + x^j + y^k

, are linearly dependent and

| \to c | = \sqrt{3}

then

(x, y)

$(1, 1)$
$(- 2, 0)$
$(\frac{1}{5}, \frac{7}{5})$
$(- \frac{7}{5}, \frac{3}{5})$

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Q. If

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

and

\to c =^i + α^j + β^k

are linearly dependent vectors and

| \to c | = \sqrt{3},

then

$α = 1, β = - 1$
$α = 1, β = \pm 1$
$α = - 1, β = \pm 1$
$α = \pm 1, β = 1$

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$\frac{\to b}{| \to b |^{2}} + \frac{\to a \times \to b}{| \to a \times \to b |^{2}}$
$\frac{\to b}{| \to b |} + \frac{\to a \times \to b}{| \to a \times \to b |^{2}}$
$\frac{\to b}{| \to b |^{2}} + \frac{\to a \times \to b}{| \to a \times \to b |}$
None of these

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