Scalar Triple...

Scalar Triple Product

Trending Questions

Q. If

[\to a \to b \to c] = - 4,

then the Volume (in cubic units) of the parallelopiped with coterminous edges

\to a + 2 \to b, 2 \to b + \to c, 3 \to c + \to a

$64$
$32$
$8$
$16$

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Q. Let

\to a, \to b, \to c

be three vectors mutually perpendicular to each other and have same magnitude. If a vector

\to r

satisfies

\to a \times {(\to r - \to b) \times \to a} + \to b \times {(\to r - \to c) \times \to b} + \to c \times {(\to r - \to a) \times \to c} = \to 0

then

\to r

is equal to

$\frac{1}{3} (\to a + \to b + \to c)$
$\frac{1}{2} (\to a + \to b + 2 \to c)$
$\frac{1}{2} (\to a + \to b + \to c)$
$\frac{1}{3} (2 \to a + \to b - \to c)$

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What is $a$ cross $b$ cross $c$ ?

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Q. Let

^a,^b

be unit vectors. If

\to c

be a vector such that the angle between

^a

and

\to c

\frac{π}{12},

and

^b = \to c + 2 (\to c \times^a)

, then

{∣ ∣ 6 \to c ∣ ∣}^{2}

is equal to

$6 (3 + \sqrt{3})$
$6 (3 - \sqrt{3})$
$3 + \sqrt{3}$
$6 (\sqrt{3} + 1)$

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Q. Two fair dice are thrown. The numbers on them are taken as

λ

and

μ,

and a system of linear equations

x + y + z = 5

x + 2 y + 3 z = μ

x + 3 y + λ z = 1

is constructed. If

p

is the probability that the system has a unique solution and

q

is the probability that the system has no solution, then

$p = \frac{1}{6} and q = \frac{1}{36}$
$p = \frac{5}{6} and q = \frac{5}{36}$
$p = \frac{1}{6} and q = \frac{5}{36}$
$p = \frac{5}{6} and q = \frac{1}{36}$

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If the vectors $c, a = x \hat{i} + y \hat{j} + z \hat{k}$ and $b = \hat{j}$ are such that $a, c,$ and $b$ form a right handed system. Then $c$ is?

$z \hat{i} - x \hat{k}$
$0$
$y \hat{j}$
$- z \hat{i} + x \hat{k}$

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Q. If

\to a, \to b, \to c

are three non coplanar, non zero vectors and

\to r

is any vector in space, then

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

is equal to

λ [\to a \to b \to c]

. Then the value of

λ

$1$
$- 1$
$2$
$- 2$

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Q. If

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}

and

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

, where

\to a, \to b

and

\to c

are three non-coplanar vectors, then the value of the expression

(\to a + \to b + \to c) \cdot (\to p + \to q + \to r)

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Q. If

\to a \to b \to c

are three non-coplanar unit vectors then

[\to a \to b \to c]

is equal to

$\pm 2$
$\pm 1$
$2$
$\pm 3$

View Solution

If $u, v$ and $w$ are the three non-coplanar vectors, then $(u + v - w) ∙ (u - v) \times (v - w)$ is equal to

$u ∙ w \times u$
$u ∙ v \times w$
$0$
$3 u ∙ v \times w$

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Q. Let vectors

\to a

and

\to b

make an angle

θ = \frac{2 π}{3}

between them. If

∣ ∣ \to a ∣ ∣ = 1

and

∣ ∣ ∣ \to b ∣ ∣ ∣ = 2,

then

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

$300$
$75$
$400$
$350$

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Q. Value of scalar triple product is zero if any 2 vectors are identical.

False
True

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Q. The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.

TRUE
FALSE

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Q. Two perpendicular unit vectors

\to a

and

\to b

are such that

[\to r \to a \to b] = \frac{5}{4},

\to r \cdot (3 \to a + 2 \to b) = 0

and

\frac{- 4}{3} \to r . \to b \int - 2 \to r . \to a \frac{x + 1}{x^{2} + 1} d x = \frac{π}{2}

. Then which of the following is(are) correct ?

$| \to r |^{2} = \frac{19}{8}$
$| \to r |^{2} = \frac{7}{4}$
$\to r = \frac{\to a}{2} - \frac{3 \to b}{4} + \frac{5}{4} (\to a \times \to b)$
$\to r = - \frac{\to a}{2} + \frac{3 \to b}{2} + \frac{5}{4} (\to a \times \to b)$

View Solution

Q. Which of the following represent(s) a vector

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

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Q. Let

\to a, \to b

and

\to c

be three non-coplanar vectors and

\to d = sin x (\to a \times \to b) + cos y (\to b \times \to c) + 2 (\to c \times \to a)

be a non-zero vector, which is perpendicular to

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

If the minimum value of

x^{2} + y^{2}

is equal to

\frac{λ π^{2}}{4},

then the value of

λ

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The volume of tetrahedron whose vertices are

A = (3, 2, 1) , ~B = (1, 2, 4), ~ C = (4, 0, 3), ~ D = (1, 1, 7)~will be $- --- -$ cubic units

5
None of the above

View Solution

A B C D

is a regular tetrahedron.

M (0, 0, 0)

is the midpoint of the altitude

D N

of the tetrahedron. Coordinates of the points

A

and

B

are

(0, 0, - 2)

and

(b, b, 0)

respectively (with

b > 0

). Also

x

- coordinate of the point

C

is given to be positive. Then which of the following is/are correct?

Image of point $D$ in the plane generated by $A B C$ is $(2 \sqrt{2}, 0, - 2)$
Image of point $D$ in the plane generated by $A B C$ is $(2, 0, - 2 \sqrt{2})$
Volume of tetrahedron $A C B M$ is $\frac{4 \sqrt{2}}{3}$
Volume of tetrahedron $A C B M$ is $\frac{4}{3}$

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Q. If

\to A, \to B, \to C

are non-zero vectors. Then the scalar

\to A \cdot (\to B + \to C) \times (\to A + \to B + \to C)

equals to:

$0$
$[\to A \to B \to C] + [\to B \to C \to A]$
$[\to A \to B \to C]$
None of these

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If $\to a$ and $\to b$ are unit vectors, then the greatest value of $\sqrt{3} ∣ ∣ \begin{matrix} \to a + \to b \end{matrix} ∣ ∣ + ∣ ∣ \begin{matrix} \to a - \to b \end{matrix} ∣ ∣$ is

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If $\to a, \to b, \to c$ are non-coplanar vectors and $\to d = λ \to a + μ \to b + ν \to c,$ then $λ$ equal to

$\frac{[\to d \to b \to c]}{[\to b \to a \to c]}$
$\frac{[\to b \to c \to d]}{[\to b \to c \to a]}$
$\frac{[\to b \to d \to c]}{[\to a \to b \to c]}$
$\frac{[\to c \to b \to d]}{[\to a \to b \to c]}$

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Q. For non-zero vectors

\to p, \to q

and

\to r,

let

(\to p \times \to q) \times \to r + (\to q \cdot \to r) \to q = (x^{2} + y^{2}) \to q + (14 - 4 x - 6 y) \to p

and

(\to r \cdot \to r) \to p = \to r

where

\to p

and

\to q

are non-collinear, and

x

and

y

are scalars. Then the value of

(x + y)

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Q. If

\frac{1}{α_{k} + i}

are 8 vertices of a regular octogon where

α_{k} ϵ R, 1, 2, 3, . . . .8

(where

i = (\sqrt{- 1})

then area of the regular octagon is:-

$1$
$\sqrt{2}$
None of these
$\frac{1}{\sqrt{2}}$

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Q. If

\to a, \to b, \to c

are three non coplanar vectors and a vector

\to α

is such that

\to α = p (\to b \times \to c) + q (\to c \times \to a) + r (\to a \times \to b) and \to α \cdot (\to a + \to b + \to c) = 1

, then [\to a \to b \to c]

is equal to

$p + q + r$
$\frac{1}{p + q + r}$
$2 (p + q + r)$
$\frac{2}{p + q + r}$

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Q. The value of

[\to a - \to b \to b - \to c \to c - \to a]

where

| \to a | = 1, | \to b | = 5, | \to c | = 3

, is

0
6
None of these
1

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Q. Let

\to u

and

\to v

be the unit vectors such that

\to u \times \to v + \to u = \to w

and

\to w \times \to u = \to v

. Then the value of

[\to u \to v \to w]

is equal to

$- 1$
$0$
$1$
$2$

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Q. If

\to a

and

\to b

are perpendicular vectors,

| \to a + \to b | = 13

and

| \to a | = 5

, find the value of

| \to b |

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Q. A function is selected randomly from the set of functions defined from set

A

to set

B

, where

n (A) = 4

and

n (B) = 7

. Then the probability that the selected function is not injection, is

$\frac{223}{343}$
$\frac{120}{343}$
$\frac{210}{343}$
$\frac{123}{343}$

View Solution

If \to a, \to b and \to c

are non-coplanar vectors and

\to a \times \to c

is perpendicular to

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

then the value of

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

is equal to

$[\to a \to b \to c] \to c$
$[\to a \to b \to c] \to b$
$\to 0$
$[\to a \to b \to c] \to a$

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Q. For any three vectors,

\to a, \to b, \to c

, the value of

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

is:

$\to b \cdot (\to c \times \to a)$
$2 \to b \cdot (\to a \times \to c)$
$0$
$2 \to a \cdot (\to b \times \to c)$

View Solution