Scalar Triple...

Scalar Triple Product

Trending Questions

If $\to a, \to b$ and $\to c$ are three non-coplanar vectors, then $(\to a + \to b + \to c) \cdot [(\to a + \to b) \times (\to a + \to c)]$ equals

0
$[\to a \to b \to c]$
$2. [\to a \to b \to c]$
$- [\to a \to b \to c]$

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$\to V = 2^i +^j -^k$ and $\to W =^i + 3^k$ . If $\to U$ is a unit vector, then the maximum value of the scalar triple product $[\to U \to V \to W]$ is

-1
$\sqrt{10} + \sqrt{6}$
$\sqrt{59}$
$\sqrt{60}$

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Q. If a, b, c and d are unit vectors such that (a x b).(c x d) = 1 and a.c =

\frac{1}{2}

then

b and d are non-parallel
b, c and d are non-coplanar
None of these
a, b and c are non-coplanar

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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

$0$
$1$
None of these
$6$

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Q. The value of a for which the vectors
A=2i-3j+3k
B=3i+aj-7k
C=5i+3j+6k
are coplanar is

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(i) If

A (z_{1}), B (z_{2})

are two complex numbers, then distance between

z_{1}

z_{2}

A B = | z_{1} - z_{2} |

(ii) Line segment joining the points

A (z_{1}), B (z_{2})

is divided by the point

P (z)

in the ratio

m : n

, then

z = \frac{m z_{2} + n z_{1}}{m + n}

where

m, n

are real numbers.

(iii) Points

z_{1}, z_{2}, z_{3}

are collinear if

∣ ∣ ∣ ∣ \begin{matrix} z_{1} & _{1} & 1 z_{2} & _{2} & 1 z_{3} & _{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0

and the general equation of line joining

A (z_{1}), B (z_{2})

in non-parametric given by

∣ ∣ ∣ ∣ \begin{matrix} z & ¯ z & 1 z_{1} & _{1} & 1 z_{2} & _{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0

and general equation of straight line is

a ¯ z + ¯ a z + b = 0

where

a

is a complex number &

b

is real, the real slope of the line is

\frac{a + ¯ a}{i (¯ a - a)}

or equal to

\frac{- R e (a)}{I m g (a)}

and the complex slope of the line is

- \frac{a}{¯ a}

\frac{coefficient of ¯ z}{coefficient of z}

.The lines

¯ a z + a ¯ z + λ = 0

and

z ¯ a - ¯ z a + i λ = 0

are respectively parallel and

⊥

to the line

a ¯ z + ¯ a z + b = 0

(iv) The length of

⊥

from a point

z_{1}

to the line

¯ a z + a ¯ z + b = 0

is given by

\frac{| a_{1} + ¯ a z_{1} + b |}{2 | a |}

(v) If the point

A (z_{1}), B (z_{2})

are to be considered as foci and length of major axis is

2 a

then equation of ellipse is given by

| z - z_{1} | + | z - z_{2} | = 2 a ⟹ 2 a > | z_{1} - z_{2} |

If major axis is to be considered as transversal axis with length

2 a

such that

2 a < | z_{1} - z_{2} |

The equation of hyperbola is given by

∥ z - z_{1} ∣ - ∣ z - z_{2} ∥ = 2 a

(vi) If

| z - α | = r

(α

is complex) is a circle at

α = (p, q)

& radius

= r

Let

α, β

be two fixed non zero complex numbers & '

z

' a variable complex number such that the lines

α ¯ z + ¯ α z + 1 = 0

and

β ¯ z + ¯ β z + 1 = 0

are mutually perpendicular lines, then value of

α ¯ β + β ¯ α

equals

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

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The volume of parallelopipped formed by following 3 vectors will be___cubic units.

$\to a = 3^i - 2^j + 5^k \to b = 2^i + 2^j -^k \to c = - 4^i + 3^j + 2^k$

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