Linear Combin...

Linear Combination of Vectors

Trending Questions

Q. Let

\to a, \to b, \to c

be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle

θ

, with the vector

\to a + \to b + \to c .

Then

36 {cos}^{2} 2 θ

is equal to

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Q. If

\to a

and

\to b

are two vectors, then the value of

(\to a - \to b) \times (\to a + \to b)

$(\to a \times \to b)$
$2 (\to a \times \to b)$
$3 (\to a + \to b)$
$\to 0$

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If a.a = 0 and a.b = 0, then what can be conclude about the vector b?

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Q. If

l o g_{6}^{30} = a, l o g_{15}^{24} = b t h e n l o g_{12}^{60} =

$\frac{2 a b}{a b + b + 1}$
$\frac{a b + b - 1}{2 a b + b}$
$\frac{2 a (b + 1)}{a b + b}$
$\frac{2 a b + 2 a - 1}{a b + b + 1}$

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Q. If

\to a

and

\to b

are non-zero and non-collinear vectors, then

[\to a \to b^i]^i + [\to a \to b^j]^j + [\to a \to b^k]^k

is equal to

$\to a + \to b$
$\to a \times \to b$
$\to a - \to b$
$\to b \times \to a$

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Q. Let

\to α, \to β, \to γ

be three unit vectors such that

\to α . \to β = \to α . \to γ = 0.

If the angle between

\to β

and

\to γ

30^{\circ},

then find

\to α

in terms of

\to β

and

\to γ

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Write down a unit vector in XY- plane, making an angle of $30^{\circ}$ in anticlockwise direction with the positive direction of X-axis.

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Q. The equation

[\begin{matrix} 1 & x & y \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 1 0 & 2 & - 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 x y \end{matrix} ⎤ ⎥ ⎦ = [0]

has

rational roots if $y = - 1$
irrational roots if $y = 0$
rational roots if $y = 0$
integral roots if $y = - 1$

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Evaluate the following.

$6 a b^{2} c - 2 a^{2} c + b^{2} c$ $f o r a = 2, b = 3 a n d c = 1$

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Q. If

\to a

is a non-zero vector, then the value of

^i \times (\to a \times^i) +^j \times (\to a \times^j) +^k \times (\to a \times^k)

$3 \to a$
$2 \to a$
$\to a$
$\to 0$

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Q. If

(x + \frac{1}{x}) = 2 \sqrt{3}

, then the value of

(x^{3} + \frac{1}{x^{3}})

$12 \sqrt{3}$
$18$
$18 \sqrt{3}$
None of these

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Q. If

\to a

and

\to b

are two non zero vectors such that

\to a \times \to b = \to b \times \to a,

then

$\to a \cdot \to b = 0$
$\to a = k \to b,$ where $k$ is a scalar quantity
$\to a$ and $\to b$ are always unlike vectors
$\to a$ and $\to b$ are always equal vectors

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Q. Let

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

and

\to c =^i + 2^j + 3^k

be three given vectors. If

\to r

is a vector such that

\to r \times \to b = \to c \times \to b

and

\to r \cdot \to a = 0,

then the value of

\to r \cdot \to b =

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Q. If

\to a, \to b, \to c, \to d

are non-zero vectors, then

[\to a \times \to b \to a \times \to c \to d]

can be simplified as:

$(\to a \cdot \to b) [\to a \to c \to d]$
$(\to a \cdot \to d) [\to a \to b \to c]$
$(\to b \cdot \to c) [\to a \to b \to d]$
$(\to a \cdot \to c) [\to b \to c \to d]$

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Q. Find

\vec{a} . (\vec{b} \times \vec{c})

, if

\vec{a} = 2 \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k}

and

\vec{c} = 3 \hat{i} + \hat{j} + 2 \hat{k}

. [CBSE 2014]

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Q. Let

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

. Then, vector

\to r

satisfying the equations

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

and

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

$^i -^j + 3^k$
$3^i -^j +^k$
$3^i +^j -^k$
$^i -^j -^k$

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Q. If

_{1} a n d_{2}

are two vectors in x – y plane. Then any vector in that plane can be obtained by the linear combination of these two vectors.The statement is

True
False

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Q. Two unit vectors

\to a

and

\to b

are pependicular to each other. Another unit vector

\to c

is inclined at an angle

α

to both

\to a

and

\to b

. If

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

, then

$x^{2} + y^{2} = 1$
$x^{2} = y^{2}$
$z^{2} = - cos 2 α$
$x^{2} + y^{2} + z^{2} = 1$

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Q. Find the magnitude of the vector

\vec{a} = 2 \hat{i} + 3 \hat{j} - 6 \hat{k} .

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4 \int 0 | x^{2} - 4 x + 3 | d x

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Q. Let,

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

A vector coplanar to

\to a

and

\to b

has a projection along

\to c

of magnitude

\frac{1}{\sqrt{3}}

, then the vector is

$4^i -^j + 4^k$
$4^i +^j - 4 k$
None of these
$2^i +^j +^k$

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Q. If

\to a, \to b, \to c

are any three non-zero vectors such that

(\to a + \to b) \cdot \to c = (\to a - \to b) \cdot ¯ ¯ c = 0,

then

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

$\to 0$
$\to a$
$\to b$
$\to c$

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For given vectors, $a = 2^i -^j + 2^k a n d b = -^i +^j -^k$ , find the unit vector in the direction of the vector $a + b$ .

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Q. If

\to a

and

\to b

are two non collinear unit vectors and

| \to a + \to b | = \sqrt{3}

, then

(2 \to a - 5 \to b) . (3 \to a + \to b) =

$\frac{13}{2}$
$\frac{11}{2}$
$0$
$- \frac{11}{2}$

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Q. Three points whose position vectors are

\to a

\to b

\to c

will be collinear if

$λ \to a + μ \to b = (λ + μ) \to c$
None of these
$\to a \times \to b + \to b \times \to c + \to c \times \to a = \to 0$
$[\begin{matrix} \to a & \to b & \to c \end{matrix}] = 0$

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Q. Let

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

and a unit vector

\to c

be coplanar. If

\to c

is perpendicular to

\to a,

then

\to c

is equal to

$\frac{1}{\sqrt{2}} (-^j +^k)$
$\frac{1}{\sqrt{3}} (-^i -^j -^k)$
$\frac{1}{\sqrt{5}} (-^i - 2^j)$
$\frac{1}{\sqrt{5}} (^i -^j -^k)$

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Q. The scalar product of

\to a and \to b

is defined as

\to a . \to b = | \to a | | \to b | cos θ, θ

is angle between them. The vector product of

\to a a n d \to b

is defined as

\to a \times \to b = | \to a | . | \to b | sin θ ˆ n

. Thus

\to a \times \to b

is a vector whose magnitude is

| \to a | | \to b | sin θ

and direction is perpendicular to the plane of

\to a and \to b

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Q. The vector product

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

is equal to:

$[\to a \to c \to b] \to b$
$[\to a \to b \to c] \to b$
$[\to a \to c \to b] \to c$
$[\to a \to b \to c] \to c$

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Q. Find the unit vectors perpendicular to both

\to a

and

\to b

, when

\to a = 3^i +^j - 2^k

and

\to b = 2^i + 3^j -^k

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Q. If

\to a \cdot \to a = 0

and

\to a \cdot \to b = 0

, then what can be concluded about the vector

\to b

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