Vector Triple...

Vector Triple Product

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Q. Let

\to a

and

\to b

be two vectors such that

∣ ∣ ∣ 2 \to a + 3 \to b ∣ ∣ ∣ = ∣ ∣ ∣ 3 \to a + \to b ∣ ∣ ∣

and the angle between

\to a

and

\to b

60^{\circ} .

\frac{1}{8} \to a

is a unit vector, then

∣ ∣ ∣ \to b ∣ ∣ ∣

is equal to:

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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. Let

\to a

be a vector which is perpendicular to thevector

3^i + \frac{1}{2}^j + 2^k .

\to a \times (2^i +^k) = 2^i - 13^j - 4^k,

then the projection of the vector

a

on the vector

2^i + 2^j +^k

is :

$1$
$\frac{7}{3}$
$\frac{1}{3}$
$\frac{5}{3}$

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Q. The unit vector which is orthogonal to the vector

3^i + 2^j + 6^k

is coplanar with vectors

2^i +^j + 6^k a n d^i -^j -^k

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Integrate $\int \frac{\sin (x)}{2 + \sin (2 x)}$ .

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Q. If the vectors

\vec{a} = 2 \hat{i} - 3 \hat{j}

and

\vec{b} = - 6 \hat{i} + m \hat{j}

are collinear, find the value of m.

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Q. Find a vector in the direction of vector

2 \hat{i} - 3 \hat{j} + 6 \hat{k}

which has magnitude 21 units. [CBSE 2014]

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Q. If

\to a, \to b, \to c

are non-coplanar vectors and a vector

\to V

satisfying

\to V \cdot \to a = \to V \cdot \to b = \to V \cdot \to c = 0,

then

| \to V |

is equal to

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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

is a perpendicular to

\to b

and

\to c, ∣ ∣ \to a ∣ ∣ = 2, ∣ ∣ ∣ \to b ∣ ∣ ∣ = 3, ∣ ∣ \to c ∣ ∣ = 4

and the angle between

\to b

and

\to c is \frac{2 π}{3}, then ∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣

is equal to

$24$
$12$
$12 \sqrt{3}$
$24 \sqrt{3}$

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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. Unit vectors

\to a, \to b, \to c

are coplanar. A unit vector

\to d

is perpendicular to them. If

(\to a \times \to b) \times (\to c \times \to d) = \frac{1}{6}^i - \frac{1}{3}^j + \frac{1}{3}^k

and the angle between

\to a

and

\to b

30^{\circ}

then

\to c

is equal to

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

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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. If three non-zero vectors are

a = a_{1} i + a_{2} j + a_{3} k, b = b_{1} i + b_{2} j + b_{3} k

and

c = c_{1} i + c_{2} j + c_{3} k

If c is the unit vector perpendicular to the vectors a and b and the angle between a and b is

\frac{π}{6}

, then

{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}

is equal to

0
1
$\frac{3 (\sum a_{1}^{2}) (\sum b_{1}^{2}) (\sum c_{1}^{2})}{4}$
$\frac{(\sum a_{1}^{2}) (\sum b_{1}^{2})}{4}$

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Q. If

\to a, \to b, \to c

are three non coplanar vectors and a vector

\to r

satisfying the vector equation

\to r . \to a = \to r . \to b = \to r . \to c = 1

is:

$\frac{1}{[\to a \to b \to c]} ((\to a \cdot \to b) + (\to b \cdot \to c) + (\to c \cdot \to a))$
$\frac{1}{[\to a \to b \to c]} ((\to a \times \to b) + (\to b \times \to c) + (\to a \times \to c))$
$\frac{(\to a \times \to b)}{[\to a \to b \to c]} + \frac{(\to c \times \to b)}{[\to a \to b \to c]} + \frac{(\to c \times \to a)}{[\to a \to b \to c]}$
$\frac{(\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a)}{[\to a \to b \to c]}$

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Q. Let

\to a, \to b, \to c

be three vectors such that

| \to a | = | \to b | = | \to c | = 4

and angle between

\to a

and

\to b

π / 3

angle between

\to b

and

\to c

π / 3

and angle between

\to c

and

\to a

π / 3.

The volume of trianglular prism whose adjacent edges are represented by the vectors

\to a, \to b

and

\to c

$12 \sqrt{2}$
$12 \sqrt{3}$
$16 \sqrt{2}$
$16 \sqrt{3}$

View Solution

If $\vec{a}$ and $\vec{b}$ are perpendicular, then $\vec{a} \times (\vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})))$ is equal to :

$(\frac{1}{2}) {|\vec{a}|}^{4} \vec{b}$
$\vec{a} \times \vec{b}$
${|\vec{a}|}^{4} \vec{b}$
$\vec{0}$

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Q. Find a vector of magnitude

\sqrt{171}

which is perpendicular to both of the vectors

\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}

and

\vec{b} = 3 \hat{i} - \hat{j} + 2 \hat{k}

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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

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

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Q. If

^i \times ((^a -^j) \times^i) +^j \times ((^a -^k) \times^j) +^k \times ((^a -^i) \times^k) = \to 0

and

\to a = x^i + y^j + z^k,

then

8 (x^{3} - x y + z x)

is equal to

View Solution

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 false
Statement $1$ is false and Statement $2$ is true

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Distance between the two planes: and is

(A)2 units (B)4 units (C)8 units

(D)

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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

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

View Solution

Q. Show the each of the following triads of vectors are coplanar:
(i)

\vec{a} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{b} = 3 \hat{i} + 2 \hat{j} + 7 \hat{k}, \vec{c} = 5 \hat{i} + 6 \hat{j} + 5 \hat{k}

(ii)

\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k}, \vec{b} = - \hat{i} + 4 \hat{j} + 3 \hat{k}, \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}

(iii)

\hat{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \hat{b} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}, \hat{c} = \hat{i} - 3 \hat{j} + 5 \hat{k}

View Solution

Q. Show that the plane whose vector equation is

\vec{r} \cdot (\hat{i} + 2 \hat{j} - \hat{k}) = 3

contains the line whose vector equation is

\vec{r} = \hat{i} + \hat{j} + λ (2 \hat{i} + \hat{j} + 4 \hat{k}) .

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Q. If the vectors

\vec{a}

= 2

\hat{i}

− 3

\hat{j}

and

\vec{b}

=−6

\hat{i}

+ m

\hat{j}

are collinear, find the value of m.

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Q. The angle between the planes

x + y + z = 0

and

3 x - 4 y + 5 z = 0

${cos}^{- 1} (\frac{1}{5} \sqrt{\frac{2}{5}})$
$\frac{π}{2}$
$\frac{π}{3}$
${cos}^{- 1} (\frac{2}{5} \sqrt{\frac{2}{3}})$

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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

^{1},^{1},^{1}

is the reciprocal system of vector triad of

\to a, \to b, \to c

, then

(\to a + \to b) \cdot^{1} + (\to b + \to c) \cdot^{1} + (\to c + \to a) \cdot^{1} =

View Solution

Q. Show that the line whose vector equation is

\vec{r} = 2 \hat{i} + 5 \hat{j} + 7 \hat{k} + λ (\hat{i} + 3 \hat{j} + 4 \hat{k})

is parallel to the plane whose vector equation is

\vec{r} \cdot (\hat{i} + \hat{j} - \hat{k}) = 7 .

Also, find the distance between them.

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