Vector Triple...

Vector Triple Product

Trending Questions

Q. If

\to a

and

\to b

are perpendicular, then

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

is equal to :

$\frac{1}{2} | \to a |^{4} \to b$
$\to a \times \to b$
$| \to a |^{4} \to b$
$\to 0$

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If a point $P(x,y)$ moves along the ellipse ${(x}^{2} {/16)+(y}^{2} /25)=1$ and $C$ is the centre of the ellipse, then the sum of maximum and minimum values of $CP$ is

$25$
$9$
$4$
$5$
$16$

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Let $^a,^b a n d^c$ be three unit vectors such that
$^a \times (^b \times^c) = \frac{\sqrt{3}}{2} (^b +^c) .$ If
$^b$ is not parallel to $^c$ , then the angle between $^a$ and $^b$ is

$\frac{3 π}{4}$
$\frac{π}{2}$
$\frac{2 π}{3}$
$\frac{5 π}{6}$

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Q. If

\to a, \to b, \to c

are non-coplanar unit vectors such that

\to a \times (\to b \times \to c) = \frac{(\to b + \to c)}{\sqrt{2}},

then the angle between

\to a

and

\to b

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

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Let $Δ P Q R$ be a triangle. Let a = QR, b = RP and c = PQ. If $| a | = 12, | b | = 4 \sqrt{3}$ and $b . c = 24,$ then which of the following is/are true ?

$\frac{| c |^{2}}{2} - | a | = 12$
$\frac{| c |^{2}}{2} + | a | = 30$
$| a \times b + c \times a | = 48 \sqrt{3}$
$a . b = - 72$

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Q. If

\to x, \to y, \to z

are units vectors such that

\to x + \to y + \to z = \to a

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

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

and

| \to a | = 2,

then which of the following is FALSE?

$\to y$ is perpendicular to $\to z$
$\to y$ is perpendicular to $\to b$
Angle between $\to x$ and $\to y$ is acute.
Angle between $\to x$ and $\to z$ is obtuse.

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Q. Let

\to x, \to y

and

\to z

be unit vectors such that

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

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

and

| \to a | = 2

.

Then which of the following option(s) is/are CORRECT ?

$\to a \cdot \to z = \frac{3}{2}$
$\to y \cdot \to z = 0$
$\to z = \frac{4}{3} (\to c - \to b)$
$\to y = 4 \to c$

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Q. If

\to a

and

\to b

are perpendicular, then

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

is equal to :

$\frac{1}{2} | \to a |^{4} \to b$
$\to a \times \to b$
$| \to a |^{4} \to b$
$\to 0$

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Q. Let

\to p, \to q

and

\to r

be three non-coplanar unit vectors equally inclined to each other at an angle of

\frac{π}{3}

. Then the value of

| \to p \times (\to q \times \to r) |

$3$
$\frac{1}{2}$
$\frac{\sqrt{3}}{4}$
$\frac{3 \sqrt{3}}{4}$

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Q. If

\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1

and

\to y \cdot \to z = 1

. Vector

\to x

$\frac{1}{| \to a \times \to b |^{2}} [\to a \times (\to a \times \to b)]$
$\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b - \to a \times (\to a \times \to b)]$
$\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b + \to b \times (\to a \times \to b)]$
None of these

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Q. If

\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1

and

\to y \cdot \to z = 1

. Vector

\to z

$\frac{γ}{| \to a \times \to b |^{2}} [\to a + \to b \times (\to a \times \to b)]$
$\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b - \to a \times (\to a \times \to b)]$
$\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b + \to b \times (\to a \times \to b)]$
None of these

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Q. If

\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1

and

\to y \cdot \to z = 1

. Vector

\to y

$\frac{\to a \times \to b}{γ}$
$\to a + \frac{\to a \times \to b}{γ}$
$\to a + \to b + \frac{\to a \times \to b}{γ}$
None of these.

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Q. The vector

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

Perpendicular to $\to b$ and parallel to $\to c$
Coplanar with $\to b a n d \to c$ and orthogonal to $\to a$
(c) Perpendicular to both $\to b a n d \to c$
None of the above

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Q. Let

\to x, \to y

and

\to z

be unit vectors such that

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

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

and

| \to a | = 2

.

Then which of the following option(s) is/are CORRECT ?

$\to a \cdot \to z = \frac{3}{2}$
$\to y \cdot \to z = 0$
$\to z = \frac{4}{3} (\to c - \to b)$
$\to y = 4 \to c$

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Q. If a, b, c are non-coplanar unit vectors such that

a \times (b \times c) = \frac{b + c}{\sqrt{2}}

, then the angle between a and b is

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

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Q. If

\to b

and

\to c

are non-zero and non-collinear vectors,

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

= (4 - 2 x - sin y) \to b + (x^{2} - 1) \to c

and

(\to c \cdot \to c) \to a = \to c

. Then,

$x = 1$
$x = - 1$
$y = (4 n + 1) \frac{π}{2}, n \in I$
$y = (2 n + 1) \frac{π}{2}, n \in I$

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Q. Statement

1

: If

\to A = 2^i + 3^j + 6^k, \to B =^i +^j - 2^k

and

\to C =^i + 2^j +^k

, then

| \to A \times (\to A \times (\to A \times \to B)) \cdot \to C | = 243

Statement

2

| \to A \times (\to A \times (\to A \times \to B)) \cdot \to C | = | \to A |^{2} | [\to A \to B \to C] |

Both the statements are true and statement $2$ is the correct explanation for Statement $1$
Both the statements are true but Statement $2$ is not the correct explanation for Statement $1$
Statement $1$ is false and Statement $2$ is true
Statement $1$ is false and Statement $2$ is false

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Q. If a, b, c are non-coplanar unit vectors such that

a \times (b \times c) = \frac{b + c}{\sqrt{2}}

, then the angle between a and b is

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

View Solution