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Cross Product...

Cross Product of Two Vectors

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Q. Let

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

and

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

be two vectors. If a vector perpendicular to both the vectors

\to a + \to b

and

\to a - \to b

has the magnitude

12

then one such vector is :

$4 (2^i + 2^j +^k)$
$4 (2^i + 2^j -^k)$
$4 (2^i - 2^j -^k)$
$4 (- 2^i - 2^j +^k)$

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Q.

If the points represented by complex numbers $z_{1}$ = $a + i b$ , $z_{2}$ = $a^{1} + i b^{1}$ and $z_{1} - z_{2}$ are coliinear, then

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Q. Let

\to a, \to b

and

\to c

be three unit vectors such that

\to a + \to b + \to c = \to 0

. If

λ = \to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a

and

\to d = \to a \times \to b + \to b \times \to c + \to c \times \to a

, then the ordered pair

(λ, \to d)

is equal to:

$(\frac{3}{2}, 3 \to a \times \to c)$
$(- \frac{3}{2}, 3 \to c \times \to b)$
$(- \frac{3}{2}, 3 \to a \times \to b)$
$(\frac{3}{2}, 3 \to b \times \to c)$

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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. Vector(s) perpendicular to both the vectors

\to A = 3^i + 5^j + 2^k

and

\to B = 2^i + 4^j + 6^k

is/are

$22^i - 14^j + 2^k$
$6^i + 20^j + 12^k$
$- (22^i - 14^j + 2^k)$
$- (6^i + 20^j + 12^k)$

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Q. Let

O

be the origin, and

- - \to O X, - - \to O Y, - - \to O Z

be three unit vectors in the directions of the sides

- - \to Q R, - - \to R P, - - \to P Q,

respectively, of a triangle

P Q R .

| - - \to O X \times - - \to O Y | =

$sin (P + Q)$
$sin 2 R$
$sin (P + R)$
$sin (Q + R)$

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Q.

\to a

and

\to b

are two vectors such that

| \to a | = 1, | \to b | = 4, | \to c |^{2} = 192

and

\to a . \to b = 2

. If

\to c = (2 \to a \times \to b) - 3 \to b

, then the angle between

\to b

and

\to c

is

$\frac{π}{3}$
$\frac{π}{6}$
$\frac{3 π}{4}$
$\frac{5 π}{6}$

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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. If

\to a, \to b,

and

\to c

are unit vectors such that

\to a + 2 \to b + 2 \to c = \to 0,

then

| \to a \times \to c |

is equal to :

$\frac{\sqrt{15}}{4}$
$\frac{1}{4}$
$\frac{15}{16}$
$\frac{\sqrt{15}}{16}$

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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. If

\to a, \to b, \to c

are non coplanar non zero vectors such that

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

and

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

then which of the following is not correct?

$| \to a | = 1$
$\to a \cdot (\to b \times \to c) = 1$
$| \to a | \neq | \to b | \neq | \to c |$
$| \to a | + | \to b | + | \to c | = 3$

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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. 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.

The angle between two vectors \to A and \to B, given that \to A . \to B = 3 and ∣ ∣ ∣ \to A \times \to B ∣ ∣ ∣ = 3 \sqrt{3} is

$30^{\circ}$
$45^{\circ}$
$60 °$

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