Vector (Cross) Product

Interactive Feed

JEE Main 2011

The vectors $a$ and $b$ are not perpendicular and $c$ and $d$ are two vectors satisfying $b \times c = b \times d$ and $a \cdot d = 0$ . Then the vector $d$ is equal to

c + (\frac{a \cdot c}{a \cdot b}) b

(B)

b + (\frac{b \cdot c}{a \cdot b}) c

(C)

c - (\frac{a \cdot c}{a \cdot b}) b

(D)

b - (\frac{b \cdot c}{a \cdot b}) c

JEE Main 2010

LEVELJEE Main

Let $a = \hat{j} - \hat{k}$ and $c = \hat{i} - \hat{j} - \hat{k}$ . Then the vector $b$ satisfying $a \times b + c = 0$ and $a \cdot b = 3$

(A)

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

(B)

\hat{i} - \hat{j} - 2 \hat{k}

(C)

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

(D)

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

JEE Main 2007

LEVELBoard

If $u$ and $v$ are unit vectors and $θ$ is the acute angle between them, then $2 u \times 3 v$ is a unit vector for

(A)

no value of

θ

(B)

exactly one value of

θ

(C)

exactly two values of

θ

(D)

more than two values of

θ

JEE Main 2005

LEVELJEE Main

For any vector $a$ , the value of $(a \times \hat{i})^{2} + (a \times \hat{j})^{2} + (a \times \hat{k})^{2}$ is equal to

(A)

3 a^{2}

(B)

a^{2}

(C)

2 a^{2}

(D)

4 a^{2}

JEE Main 2003

LEVELJEE Main

Let $u = \hat{i} + \hat{j}, v = \hat{i} - \hat{j}$ and $w = \hat{i} + 2 \hat{j} + 3 \hat{k}$ . If $\overset{n}{^}$ is a unit vector such that $u \cdot \overset{n}{^} = 0$ and $v \cdot \overset{n}{^} = 0$ , then $∣ w \cdot \overset{n}{^} ∣$ is equal to

A couple is of moment $G$ and the force forming the couple is $p$ . If $p$ is turned through a right angle the moment of the couple thus formed is $H$ . If instead, the force $p$ are turned through an angle $α$ , then the moment of couple becomes

(A)

H sin α - G cos α

(B)

G sin α - H cos α

(C)

H sin α + G cos α

(D)

G sin α + H cos α

JEE Main 2002

LEVELBoard

If $∣ a ∣ = 4, ∣ b ∣ = 2$ and the angle between $a$ and $b$ is $π /6$ then $(a \times b)^{2}$ is equal to

(A)

(B)

(C)

a

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 :

(A)

z \hat{i} - x \hat{k}

(B)

0

(C)

y \hat{j}

(D)

- z \hat{i} + x \hat{k}

JEE Main 2002

LEVELBoard

$a = 3 \hat{i} - 5 \hat{j}$ and $b = 6 \hat{i} + 3 \hat{j}$ are two vectors and $c$ is a vector such that $c = a \times b$ then $∣ a ∣ : ∣ b ∣ : ∣ c ∣$

(A)

34 : 45 : 39

(B)

34 : 45 : 39

(C)

34 : 39 : 45

(D)

39 : 35 : 34

JEE Main 2002

LEVELJEE Main

If $a \times b = b \times c = c \times a$ then $a + b + c =$

(A)

a b c

(B)

- 1

(C)

0

(D)

2

JEE Advanced 2016

LEVELJEE Main

Let $\overset{u}{^} = u_{1} \hat{i} + u_{2} \hat{j} + u_{3} \hat{k}$ be a unit vector in $R^{3}$ and $\overset{w}{^} = \frac{1}{6} (\hat{i} + \hat{j} + 2 \hat{k})$ . Given that there exists a vector $v$ in $R^{3}$ such that $∣ \overset{u}{^} \times v ∣ = 1$ and $\overset{w}{^} \cdot (\overset{u}{^} \times v) = 1$ . Which of the following statement(s) is (are) correct?

* Multiple Correct Options

(A)

There is exactly one choice for such

v

(B)

There are infinitely many choices for such

v

(C)

\overset{u}{^}

lies in the xy-plane then

∣ u_{1} ∣ = ∣ u_{2} ∣

(D)

\overset{u}{^}

lies in the xz-plane then

2∣ u_{1} ∣ = ∣ u_{3} ∣

JEE Advanced 2015

LEVELJEE Main

Let $Δ P QR$ be a triangle. Let $a = QR, b = R P$ and $c = P Q$ . If $∣ a ∣ = 12, ∣ b ∣ = 43, b \cdot c = 24$ , then which of the following is (are) true?

* Multiple Correct Options

(A)

\frac{∣ c ∣ ^{2}}{2} - ∣ a ∣ = 12

(B)

\frac{∣ c ∣ ^{2}}{2} + ∣ a ∣ = 30

(C)

∣ a \times b + c \times a ∣ = 483

(D)

a \cdot b = - 72

JEE Advanced 2012

LEVELJEE Main

If $a$ and $b$ are vectors such that $∣ a + b ∣ = 29$ and $a \times (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \times b$ , then a possible value of $(a + b) \cdot (- 7 \hat{i} + 2 \hat{j} + 3 \hat{k})$ is

If $a, b, c$ and $d$ are unit vectors such that $(a \times b) \cdot (c \times d) = 1$ and $a \cdot c = \frac{1}{2}$ , then

(A)

a, b, c

are non-coplanar

(B)

b, c, d

are non-coplanar

(C)

b, d

are non-parallel

(D)

a, d

are parallel and

b, c

are parallel

JEE Advanced 2007

LEVELJEE Main

Let $a, b, c$ be unit vectors such that $a + b + c = 0$ . Which one of the following is correct?

(A)

a \times b = b \times c = c \times a = 0

(B)

a \times b = b \times c = c \times a \neq = 0

(C)

a \times b = b \times c = a \times c \neq = 0

(D)

a \times b, b \times c, c \times a

are mutually perpendicular

JEE Advanced 2007

LEVELJEE Main

Let the vectors $P Q, QR, R S, S T, T U$ and $U P$ represent the sides of a regular hexagon. STATEMENT-1 : $P Q \times (R S + S T) \neq = 0$ . because STATEMENT-2 : $P Q \times R S = 0$ and $P Q \times S T \neq = 0$ .

(A)

Statement-1 is True, statement-2 is True; Statement-2 is a correct explanation for Statement-1.

(B)

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C)

Statement-1 is True, Statement-2 is False

(D)

Statement-1 is False, Statement-2 is True.

JEE Advanced 2000S

LEVELJEE Main

If the vectors $a, b$ and $c$ form the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$ , then

(A)

a \cdot b + b \cdot c + c \cdot a = 0

(B)

a \times b = b \times c = c \times a

(C)

a \cdot b = b \cdot c = c \cdot a

(D)

a \times b + b \times c + c \times a = 0

JEE Advanced 2000S

LEVELJEE Main

Let the vectors $a, b, c$ and $d$ be such that $(a \times b) \times (c \times d) = 0$ . Let $P_{1}$ and $P_{2}$ be planes determined by the pairs of vectors $a, b$ and $c, d$ respectively. Then the angle between $P_{1}$ and $P_{2}$ is

(A)

0

(B)

\frac{π}{4}

(C)

\frac{π}{3}

(D)

\frac{π}{2}

JEE Advanced 1999

LEVELJEE Main

Let $a = 2 \hat{i} + \hat{j} - 2 \hat{k}$ and $b = \hat{i} + \hat{j}$ . If $c$ is a vector such that $a \cdot c = ∣ c ∣, ∣ c - a ∣ = 22$ and the angle between $(a \times b)$ and $c$ is $3 0^{\circ}$ , then $∣ (a \times b) \times c ∣ =$

(A)

2/3

(B)

3/2

(C)

2

(D)

3

JEE Advanced 1999

LEVELJEE Main

Let $a$ and $b$ be two non-collinear unit vectors. If $u = a - (a \cdot b) b$ and $v = a \times b$ , then $∣ v ∣$ is

* Multiple Correct Options

(A)

∣ u ∣

(B)

∣ u ∣ + ∣ u \cdot a ∣

(C)

∣ u ∣ + ∣ u \cdot b ∣

(D)

∣ u ∣ + u \cdot (a + b)

JEE Advanced 1987

LEVELJEE Main

The number of vectors of unit length perpendicular to vectors $a = (1, 1, 0)$ and $b = (0, 1, 1)$ is

Let $a = - \hat{i} - \hat{k}$ , $b = - \hat{i} + \hat{j}$ and $c = \hat{i} + 2 \hat{j} + 3 \hat{k}$ be three given vectors. If $r$ is a vector such that $r \times b = c \times b$ and $r \cdot a = 0$ , then the value of $r \cdot b$ is .........

JEE Advanced 1997

LEVELJEE Main

Let $O A = a$ , $O B = 10 a + 2 b$ and $O C = b$ where $O, A$ and $C$ are non-collinear points. Let $p$ denote the area of the quadrilateral $O A B C$ , and let $q$ denote the area of the parallelogram with $O A$ and $O C$ as adjacent sides. If $p = k q$ , then $k = .........$

JEE Advanced 1994

LEVELJEE Main

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 .........

JEE Advanced 1991

LEVELJEE Main

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The vectors a and b are not perpendicular and c and d are two vectors satisfying b×c=b×d and a⋅d=0. Then the vector d is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=j^​−k^ and c=i^−j^​−k^. Then the vector b satisfying a×b+c=0 and a⋅b=3

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If u and v are unit vectors and θ is the acute angle between them, then 2u×3v is a unit vector for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any vector a, the value of (a×i^)2+(a×j^​)2+(a×k^)2 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let u=i^+j^​,v=i^−j^​ and w=i^+2j^​+3k^. If n^ is a unit vector such that u⋅n^=0 and v⋅n^=0, then ∣w⋅n^∣ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣a∣=4,∣b∣=2 and the angle between a and b is π/6 then (a×b)2 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors c,a=xi^+yj^​+zk^ and b=j^​ are such that a,c and b form a right handed system then c is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a=3i^−5j^​ and b=6i^+3j^​ are two vectors and c is a vector such that c=a×b then ∣a∣:∣b∣:∣c∣

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a×b=b×c=c×a then a+b+c=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ΔPQR be a triangle. Let a=QR​,b=RP and c=PQ​. If ∣a∣=12,∣b∣=43​,b⋅c=24, then which of the following is (are) true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a and b are vectors such that ∣a+b∣=29​ and a×(2i^+3j^​+4k^)=(2i^+3j^​+4k^)×b, then a possible value of (a+b)⋅(−7i^+2j^​+3k^) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c and d are unit vectors such that (a×b)⋅(c×d)=1 and a⋅c=21​, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be unit vectors such that a+b+c=0. Which one of the following is correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the vectors PQ​,QR​,RS,ST,TU and UP represent the sides of a regular hexagon. STATEMENT-1 : PQ​×(RS+ST)=0. because STATEMENT-2 : PQ​×RS=0 and PQ​×ST=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors a,b and c form the sides BC,CA and AB respectively of a triangle ABC, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=2i^+j^​−2k^ and b=i^+j^​. If c is a vector such that a⋅c=∣c∣,∣c−a∣=22​ and the angle between (a×b) and c is 30∘, then ∣(a×b)×c∣=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a and b be two non-collinear unit vectors. If u=a−(a⋅b)b and v=a×b, then ∣v∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of vectors of unit length perpendicular to vectors a=(1,1,0) and b=(0,1,1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=−i^−k^, b=−i^+j^​ and c=i^+2j^​+3k^ be three given vectors. If r is a vector such that r×b=c×b and r⋅a=0, then the value of r⋅b is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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 .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given that a=(1,1,1), c=(0,1,−1), a⋅b=3 and a×b=c, then b=.........

The vectors $a$ and $b$ are not perpendicular and $c$ and $d$ are two vectors satisfying $b \times c = b \times d$ and $a \cdot d = 0$ . Then the vector $d$ is equal to

Let $a = \hat{j} - \hat{k}$ and $c = \hat{i} - \hat{j} - \hat{k}$ . Then the vector $b$ satisfying $a \times b + c = 0$ and $a \cdot b = 3$

If $u$ and $v$ are unit vectors and $θ$ is the acute angle between them, then $2 u \times 3 v$ is a unit vector for

For any vector $a$ , the value of $(a \times \hat{i})^{2} + (a \times \hat{j})^{2} + (a \times \hat{k})^{2}$ is equal to

Let $u = \hat{i} + \hat{j}, v = \hat{i} - \hat{j}$ and $w = \hat{i} + 2 \hat{j} + 3 \hat{k}$ . If $\overset{n}{^}$ is a unit vector such that $u \cdot \overset{n}{^} = 0$ and $v \cdot \overset{n}{^} = 0$ , then $∣ w \cdot \overset{n}{^} ∣$ is equal to

If $∣ a ∣ = 4, ∣ b ∣ = 2$ and the angle between $a$ and $b$ is $π /6$ then $(a \times b)^{2}$ is equal to

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 :

$a = 3 \hat{i} - 5 \hat{j}$ and $b = 6 \hat{i} + 3 \hat{j}$ are two vectors and $c$ is a vector such that $c = a \times b$ then $∣ a ∣ : ∣ b ∣ : ∣ c ∣$

If $a \times b = b \times c = c \times a$ then $a + b + c =$

Let $Δ P QR$ be a triangle. Let $a = QR, b = R P$ and $c = P Q$ . If $∣ a ∣ = 12, ∣ b ∣ = 43, b \cdot c = 24$ , then which of the following is (are) true?

If $a$ and $b$ are vectors such that $∣ a + b ∣ = 29$ and $a \times (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \times b$ , then a possible value of $(a + b) \cdot (- 7 \hat{i} + 2 \hat{j} + 3 \hat{k})$ is

If $a, b, c$ and $d$ are unit vectors such that $(a \times b) \cdot (c \times d) = 1$ and $a \cdot c = \frac{1}{2}$ , then

Let $a, b, c$ be unit vectors such that $a + b + c = 0$ . Which one of the following is correct?

Let the vectors $P Q, QR, R S, S T, T U$ and $U P$ represent the sides of a regular hexagon. STATEMENT-1 : $P Q \times (R S + S T) \neq = 0$ . because STATEMENT-2 : $P Q \times R S = 0$ and $P Q \times S T \neq = 0$ .

If the vectors $a, b$ and $c$ form the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$ , then

Let $a = 2 \hat{i} + \hat{j} - 2 \hat{k}$ and $b = \hat{i} + \hat{j}$ . If $c$ is a vector such that $a \cdot c = ∣ c ∣, ∣ c - a ∣ = 22$ and the angle between $(a \times b)$ and $c$ is $3 0^{\circ}$ , then $∣ (a \times b) \times c ∣ =$

Let $a$ and $b$ be two non-collinear unit vectors. If $u = a - (a \cdot b) b$ and $v = a \times b$ , then $∣ v ∣$ is

The number of vectors of unit length perpendicular to vectors $a = (1, 1, 0)$ and $b = (0, 1, 1)$ is

Let $a = - \hat{i} - \hat{k}$ , $b = - \hat{i} + \hat{j}$ and $c = \hat{i} + 2 \hat{j} + 3 \hat{k}$ be three given vectors. If $r$ is a vector such that $r \times b = c \times b$ and $r \cdot a = 0$ , then the value of $r \cdot b$ is .........

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 .........

Given that $a = (1, 1, 1)$ , $c = (0, 1, - 1)$ , $a \cdot b = 3$ and $a \times b = c$ , then $b = .........$