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Cross Product...

Cross Product of Two Vectors

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Q.

Let $a = 2 \hat{i} + \hat{j} - 2 \hat{k}$ and $b = \begin{array}{rcl} \hat{i} \end{array} + \begin{array}{rcl} \hat{j} \end{array}$ . If $c$ is a vector such that $a . c = |c|$ , $|c - a| = 2 \sqrt{2}$ and the angle between $a \times b$ and $c$ is the $30 °$ , then $|(a \times b) \times c|$ is equal to

$\frac{2}{3}$
$\frac{3}{2}$
$2$
$3$

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Q.

The point of intersection of the lines $\to r \times \to a = \to b \times \to a$ and $\to r \times \to b = \to a \times \to b$ is

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

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Q.

The scalars l and m such that $l \to a + m \to b = \to c$ where $\to a, \to b$ and $\to c$ are given vectors, are equal to

$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |^{2}}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |^{2}}$
$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |}$
$l = \frac{| \to c \times \to b |^{2}}{| \to a \times \to b |^{2}}$ ; $m = \frac{| \to c \times \to a |^{2}}{| \to b \times \to a |^{2}}$
$l = \frac{| \to c \times \to a |^{2}}{| \to b \times \to a |^{2}}$ ; $m = \frac{| \to c \times \to b |^{2}}{| \to a \times \to b |^{2}}$

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Q. If |a|=1, |b|=2 and the angle between a and b is

120^{\circ}

, then

{((a + 3 b) \times (3 a - b))}^{2}

=

425
375
325
300

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Q. A unit vector perpendicular to the plane of

a = 2^i - 6^j - 3^k, b = 4^i + 3^j -^k

is

$\frac{4^i + 3^j -^k}{\sqrt{26}}$
$\frac{2^i - 6^j - 3^k}{7}$
$\frac{3^i - 2^j + 6^k}{7}$
$\frac{2^i - 3^j - 6^k}{7}$

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Q. Let

\to a = 2^i +^j - 2^k a n d \to b =^i +^j .

If

\to c

is a vector such that

\to a . \to c = | \to c |, | \to c - \to a | = 2 \sqrt{2}

and the angle between

(\to a \times \to b) a n d \to c i s 30^{\circ}, t h e n | (\to a \times \to b) \times \to c |

is equal to

$2$
$3$
$\frac{2}{3}$
$\frac{3}{2}$

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Q. The number of all possible triplets

(a_{1}, a_{2}, a_{3})

such that

a_{1} + a_{2} cos 2 x + a_{3} {sin}^{2} x = 0

for all

x

is/are

one
zero
three
none of these
infinite

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Q. If 3 vectors

¯ ¯ ¯ a . ¯ ¯ b, ¯ ¯ c

all lie in one plane (i.e., they are coplanar) then

¯ ¯ c

.

(¯ ¯ ¯ a \times ¯ ¯ b)

= _______

___

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Q. Vectors

u

and

v

are given by

u = (2, 0)

and

v = (- 3, 1)

. What is the length of vector w given by

w = - u - 2 v

?

$10$
$6$
$\sqrt{26}$
$2 \sqrt{5}$
$2$

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Q. Let

\to a = 2^i +^j - 2^k a n d \to b =^i +^j .

If

\to c

is a vector such that

\to a . \to c = | \to c |, | \to c - \to a | = 2 \sqrt{2}

and the angle between

(\to a \times \to b) a n d \to c i s 30^{\circ}, t h e n | (\to a \times \to b) \times \to c |

is equal to

$\frac{2}{3}$
$\frac{3}{2}$
$2$
$3$

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Q. Sum of coefficients of

ˆ i, ˆ j

and

ˆ k

in the cross product

(2 ˆ i + 3 ˆ j + 4 ˆ k)

\times

(ˆ i - ˆ j + ˆ k)

will be___.

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Q. Vectors

u

and

v

are given by

u = (3, 0)

and

v (1. - 4)

. What is the length of vector

w

, given by

w = 2 u - v

?

$2 \sqrt{10}$
$\sqrt{41}$
$6 - \sqrt{17}$
$3$
$\sqrt{23}$

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Q.

If $\to a \times \to b = \to c, \to b \times \to c = \to a$ and a, b, c be the moduli of the vectors $\to a, \to b, \to c$ respectively, then

a=1 , b=1
c = 1, a = 1
$\to a . (\to b \times \to c) = - 1$
b = 1, c = a

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Q.

$I f \to u = \to a - \to b a n d \to v = \to a + \to b a n d | \to a | = | \to b | = 2, t h e n | \to u \times \to v | =$

$2 \sqrt{16 - (\to a . \to b)^{2}}$
$\sqrt{16 - (\to a . \to b)^{2}}$
$2 \sqrt{4 - (\to a . \to b)^{2}}$
$\sqrt{4 - (\to a . \to b)^{2}}$

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Q.

The scalars l and m such that $l \to a + m \to b = \to c$ where $\to a, \to b$ and $\to c$ are given vectors, are equal to

$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |^{2}}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |^{2}}$
$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |}$
$l = \frac{| \to c \times \to b |^{2}}{| \to a \times \to b |^{2}}$ ; $m = \frac{| \to c \times \to a |^{2}}{| \to b \times \to a |^{2}}$
$l = \frac{| \to c \times \to a |^{2}}{| \to b \times \to a |^{2}}$ ; $m = \frac{| \to c \times \to b |^{2}}{| \to a \times \to b |^{2}}$

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Q. A unit vector perpendicular to the plane determined by the points P(1, -1, 2), Q(2, 0, -1) and R(0, 2, 1) is

$\frac{2^i +^j +^k}{\sqrt{6}}$
$\frac{2^i +^j +^k}{3}$
$\frac{2^i -^j -^k}{\sqrt{3}}$
$\frac{2^i -^j -^k}{3}$

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Q.

\to a \neq 0, \to a \times \to b = \to a \times \to c \Rightarrow \to b = \to c

True
False

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Q. Let U = {0, 1, 2 , 3, 4, 5, 6, 7}, A = {1, 3, 5, 7} and B = {0, 2, 3, 5, 7}, find the following sets.

(i) A’
(ii) B’
(iii) A ‘

\cup

B’
(iv) A’

\cap

B’
(v) (A

\cup

B)’
(vi) (A

\cap

B)’
(vii) (A’)’
(viii) (B’)’

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