Linear Depend...

Linear Dependence and Independence of Vectors

Trending Questions

Q. In a triangle

A B C

, if

| - - \to B C | = 3, | - - \to C A | = 5

and

| - - \to B A | = 7,

then the projection of the vector

- - \to B A

- - \to B C

is equal to:

$\frac{13}{2}$
$\frac{19}{2}$
$\frac{15}{2}$
$\frac{11}{2}$

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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. If

\to b and \to c

are two non-collinear unit vectors and

\to a

is any vector, then

(\to a \cdot \to b) \to b + (\to a \cdot \to c) \to c + \frac{\to a \cdot (\to b \times \to c)}{| \to b \times \to c |^{2}} (\to b \times \to c) =

$\to a$
$\to b$
$\to c$
$\to a + \to b + \to c$

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If $\to a$ , $\to b$ and $\to c$ are three vectors of equal magnitude. The angle between each pair of vectors is $\frac{π}{3}$ such that $∣ ∣ ∣ \to a + \to b + \to c ∣ ∣ ∣ = \sqrt{6}$ . Then | $\to a$ | is equal to

$2$
$- 1$
$1$
$\sqrt{6}$

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Q. The value of a for which the vectors
A=2i-j-4k ,
B=i-5j+ak and
C=2i+3j+4k are linearly dependent is

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Answer the following as true or false.

\to a

and

- \to a

are collinear

Data insufficient
True
False
Ambiguous

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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. In a triangle

A B C

, if

| - - \to B C | = 3, | - - \to C A | = 5

and

| - - \to B A | = 7,

then the projection of the vector

- - \to B A

- - \to B C

is equal to:

$\frac{11}{2}$
$\frac{19}{2}$
$\frac{13}{2}$
$\frac{15}{2}$

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Q. Vector

\to x

is Vectors

\to x, \to y

and

\to z,

each of magnitude

\sqrt{2},

make an angle of

60^{\circ}

with each other.

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

and

\to x \times \to y = \to c .

$\frac{1}{2} [(\to a - \to b) \times \to c + (\to a + \to b)]$
$\frac{1}{2} [(\to a + \to b) \times \to c + (\to a - \to b)]$
$\frac{1}{2} [- (\to a + \to b) \times \to c + (\to a + \to b)]$
$\frac{1}{2} [(\to a + \to b) \times \to c - (\to a + \to b)]$

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Q. Minimise Z=5x+4y Subject to constraints: 80x+100y>=88 40x+30y>=36 x, y>=0

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Q. Prove that the following vectors are coplanar:
(i)

2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k} and 3 \hat{i} - 4 \hat{j} - 4 \hat{k}

(ii)

\hat{i} + \hat{j} + \hat{k}, 2 \hat{i} + 3 \hat{j} - \hat{k} and - \hat{i} - 2 \hat{j} + 2 \hat{k}

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Q. Two vectors

\to a

and

\to b

are perpendicular to each other. A vector

\to c

is inclined at an angle

θ

to both

\to a

and

\to b

. If

| \to a | = 2

| \to b | = 3

| \to c | = 2

and

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

, then

$p^{2} = {cos}^{2} θ$
$r^{2} = \frac{- cos 2 θ}{9}$
$r^{2} = \frac{- {cos}^{2} θ}{9}$
$q^{2} = \frac{4 {cos}^{2} θ}{9}$

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Q. Answer the following as true or false.
(i)

\to a

and

- \to a

are collinear
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.

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Q. In a triangle

A B C

, if

| - - \to B C | = 3, | - - \to C A | = 5

and

| - - \to B A | = 7,

then the projection of the vector

- - \to B A

- - \to B C

is equal to:

$\frac{13}{2}$
$\frac{15}{2}$
$\frac{11}{2}$
$\frac{19}{2}$

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Q. Find the values of a, b, c and d from the following equations:

[\begin{matrix} 2 a + b & a - 2 b \\ 5 c - d & 4 c + 3 d \end{matrix}] = [\begin{matrix} 4 & - 3 \\ 11 & 24 \end{matrix}]

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Q. The solution of x² + y²

\frac{d y}{d x}

= 4, is
(a) x² + y² = 12x + C
(b) x² + y² = 3x + C
(c) x³ + y³ = 3x + C
(d) x³ + y³ = 12x + C

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Q. If

\to a =^i +^j +^k,^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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Is the value of the determinant different along different r o w s and columns can we evaluate one's column in the area of the triangle and get the same answer?

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Q. The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?
How far is the foot from the wall when it and the top are moving at the same rate?

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Q. If

^a

^b

are unit vectors such that

^a +^b

is also a unit vector then the angle between the vectors

^a

and

^b

$\frac{π}{6}$
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{2 π}{3}$

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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) is

(-1, -1)
(2, 3)
(3, 1)

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If $\to a, \to b, \to c$ are three vectors such that $| \to a | = 5, | \to b | = 12$ and $| \to c | = 13$ and $\to a + \to b + \to c = 0$

Find the value of $\to a . \to b + \to b . \to c + \to c . \to a$ .

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Q. If

\vec{a}

\vec{b}

\vec{c}

are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:
(i)

5 \vec{a} + 6 \vec{b} + 7 \vec{c,} 7 \vec{a} - 8 \vec{b} + 9 \vec{c} and 3 \vec{a} + 20 \vec{b} + 5 \vec{c}

(ii)

\vec{a} - 2 \vec{b} + 3 \vec{c}, - 3 \vec{b} + 5 \vec{c} and - 2 \vec{a} + 3 \vec{b} - 4 \vec{c}

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Q. The integral value of

λ

for which vectors

λ^{2}^i - 2^j, 2 λ^{2}^j + 64^k, - λ^{2}^k -^i

are linearly dependent, is equal to

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Q. The complex number

z_{1}, z_{2}

and the origin form an equilateral triangle only if

\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}} = 1

True
False

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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. Two non zero vectors

_{1} and_{2}

which are orthogonal to each other will always be linearly

dependent
independent
$either dependent or independent$

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Q. Let

¯ a = 2 ¯ i - 3 ¯ j +^k, ¯ b =^i -^j + 3^k a n d ¯ c =^i +^j -^k

Then the system of vectors

¯ a, ¯ b a n d ¯ c

Cannot be determined
None of the above
Is linearly dependent
Is linearly independent

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Q. Two non zero vectors

_{1} and_{2}

which are orthogonal to each other will always be linearly

dependent
independent
$either dependent or independent$

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