Cross Product...

Cross Product of Two Vectors

Trending Questions

Q. Let

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

and

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

, for some real

x

. Then

| \to a \times \to b | = r

is possible if :

$0 < r \leq \sqrt{\frac{3}{2}}$
$\sqrt{\frac{3}{2}} < r \leq 3 \sqrt{\frac{3}{2}}$
$3 \sqrt{\frac{3}{2}} < r < 5 \sqrt{\frac{3}{2}}$
$r \geq 5 \sqrt{\frac{3}{2}}$

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If $a$ and $b$ are unit vectors, then the vector $(a + b) \times (a \times b)$ is parallel to the vector?

$a - b$
$a + b$
$2 a - b$
$2 a + b$

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A vector perpendicular to the plane containing the points $A (1, - 1, 2)$ , $B (2, 0, - 1)$ and $C (0, 2, 1)$ is

$4 i + 8 j - 4 k$
$8 i + 4 j + 4 k$
$3 i + j + 2 k$
$i + j - k$

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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. 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. Let

\to a

and

\to b

be two vectors of equal magnitude

5

uints. Let

\to p, \to q

be vectors such that

\to p = \to a - \to b

and

\to q = \to a + \to b

. If

| \to p \times \to q | = 2 {γ - (\to a . \to b)^{2}}^{1 / 2}

, then the value of

γ

$5$
$25$
$125$
$625$

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Q. If 3 vectors

¯ ¯ ¯ a . ¯ ¯ b, ¯ ¯ c

all lie in one plane (i.e., they are coplanar) then

¯ ¯ c

(¯ ¯ ¯ a \times ¯ ¯ b)

= _______

___

View Solution

$I f \to u = \to a - \to b a n d \to v = \to a + \to b a n d | \to a | = | \to b | = 2, t h e n | \to u \times \to v | =$

$2 \sqrt{16 - (\to a . \to b)^{2}}$
$\sqrt{16 - (\to a . \to b)^{2}}$
$2 \sqrt{4 - (\to a . \to b)^{2}}$
$\sqrt{4 - (\to a . \to b)^{2}}$

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The projection of the line segment joining the points $(1, - 1, 3)$ and $(2, - 4, 11)$ on the line joining the points $(- 1, 2, 3)$ and $(3, - 2, 10)$ is

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Q. Let

\to a = 2^i +^j - 2^k a n d \to b =^i +^j .

\to c

is a vector such that

\to a . \to c = | \to c |, | \to c - \to a | = 2 \sqrt{2}

and the angle between

(\to a \times \to b) a n d \to c i s 30^{\circ}, t h e n | (\to a \times \to b) \times \to c |

is equal to

$\frac{2}{3}$
$\frac{3}{2}$
$2$
$3$

View Solution

\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

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

View Solution

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}

View Solution

Q. Let

\to a, \to b

and

\to c

be three vectors such that

| \to a | = \sqrt{3}, | \to b | = 5, \to b . \to c = 10

and the angle between

\to b

and

\to c

\frac{π}{3} .

\to a

is perpendicular to vector

\to b \times \to c

, then

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

is equal to

View Solution

The point of intersection of the lines $\to r \times \to a = \to b \times \to a$ and $\to r \times \to b = \to a \times \to b$ is

$\to a$
$\to b$
$(\to a + \to b)$
$(\to a - \to b)$

View Solution

Q. The vector perpendicular to both

\to A

and

\to B

can be calculated by taking dot product of both the vectors

False
True

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Q. Let

\to a

and

\to b

be two unit vectors such that

| \to a + \to b | = \sqrt{3} .

\to c = \to a + 2 \to b + 3 (\to a \times \to b),

then

2 | \to c |

is equal to:

$\sqrt{55}$
$\sqrt{51}$
$\sqrt{43}$
$\sqrt{37}$

View Solution

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?

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

View Solution

Q. A unit vector perpendicular to the plane determined by the points P(1, -1, 2), Q(2, 0, -1) and R(0, 2, 1) is

$\frac{2^i +^j +^k}{\sqrt{6}}$
$\frac{2^i +^j +^k}{3}$
$\frac{2^i -^j -^k}{\sqrt{3}}$
$\frac{2^i -^j -^k}{3}$

View Solution

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
$| O X \times O Y | =$

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

View Solution

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 :

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

View Solution

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 | =

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

View Solution

Q. If

a = i - j + k, a \cdot b = 0, a \times b = c,

, where

c = - 2 i - j + k

, then

b

is equal to

$(1, 0, - 1)$
$(0, 1, 1)$
$(- 1, - 1, 0)$
$(- 1, 0, 1)$

View Solution

Q. Let

^u = u_{1}^i + u_{2}^j + u_{3}^k

be a unit vector in

R^{3}

and

^w = \frac{1}{\sqrt{6}} (^i +^j + 2^k) .

Given that there exists a vector

\to v

R^{3}

such that

|^u \times \to v | = 1

and

^w \cdot (^u \times \to v) = 1.

Which of the following statement(s) is(are) correct ?

There is exactly one choice for such $\to v$
There are infinitely many choices for such $\to v$
If $^u$ lies in the $x y$ -plane then $| u_{1} | = | u_{2} |$
If $^u$ lies in the $x z$ -plane then $2 | u_{1} | = | u_{3} |$

View Solution

Q. A unit vector perpendicular to the plane of

a = 2^i - 6^j - 3^k, b = 4^i + 3^j -^k

$\frac{4^i + 3^j -^k}{\sqrt{26}}$
$\frac{2^i - 6^j - 3^k}{7}$
$\frac{3^i - 2^j + 6^k}{7}$
$\frac{2^i - 3^j - 6^k}{7}$

View Solution

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}

View Solution

Q. Let

\to a = 2^i +^j - 2^k a n d \to b =^i +^j .

\to c

is a vector such that

\to a . \to c = | \to c |, | \to c - \to a | = 2 \sqrt{2}

and the angle between

(\to a \times \to b) a n d \to c i s 30^{\circ}, t h e n | (\to a \times \to b) \times \to c |

is equal to

$\frac{2}{3}$
$\frac{3}{2}$
$2$
$3$

View Solution

Q. Sum of coefficients of

ˆ i, ˆ j

and

ˆ k

in the cross product

(2 ˆ i + 3 ˆ j + 4 ˆ k)

\times

(ˆ i - ˆ j + ˆ k)

will be___.

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Q. The vector perpendicular to both

\to A

and

\to B

can be calculated by taking dot product of both the vectors

False
True

View Solution

Q. If

a = i - j + k, a \cdot b = 0, a \times b = c,

, where

c = - 2 i - j + k

, then

b

is equal to

$(1, 0, - 1)$
$(0, 1, 1)$
$(- 1, - 1, 0)$
$(- 1, 0, 1)$

View Solution

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

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

View Solution