Vector Triple Product

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JEE Main 2016

Let $a, b$ and $c$ be three unit vectors such that $a \times (b \times c) = \frac{3}{2} (b + c)$ . If $b$ is not parallel to $c$ , then the angle between $a$ and $b$ is:

\frac{2 π}{3}

(B)

\frac{5 π}{6}

(C)

\frac{3 π}{4}

(D)

\frac{π}{2}

JEE Main 2015

LEVELJEE Main

Let $a, b$ and $c$ be three non-zero vectors such that no two of them are collinear and $(a \times b) \times c = \frac{1}{3} ∣ b ∣∣ c ∣ a$ . If $θ$ is the angle between vectors $b$ and $c$ , then a value of $sin θ$ is :

(A)

\frac{2}{3}

(B)

\frac{- 2 3}{3}

(C)

\frac{2 2}{3}

(D)

\frac{- 2}{3}

JEE Main 2011

LEVELJEE Main

If $a = \frac{1}{10} (3 \hat{i} + \hat{k})$ and $b = \frac{1}{7} (2 \hat{i} + 3 \hat{j} - 6 \hat{k})$ , then the value of $(2 a - b) \cdot [(a \times b) \times (a + 2 b)]$ is

(A)

- 3

(B)

5

(C)

3

(D)

- 5

JEE Main 2006

LEVELJEE Main

If $(a \times b) \times c = a \times (b \times c)$ where $a, b$ and $c$ are any three vectors such that $a \cdot b \neq = 0, b \cdot c \neq = 0$ then $a$ and $c$ are

(A)

inclined at an angle of

π /3

between them

(B)

inclined at an angle of

π /6

Let $a, b$ and $c$ be non-zero vectors such that $(a \times b) \times c = \frac{1}{3} ∣ b ∣∣ c ∣ a$ . If $θ$ is the acute angle between the vectors $b$ and $c$ , then $sin θ$ equals

(A)

\frac{2 2}{3}

(B)

\frac{2}{3}

(C)

\frac{2}{3}

(D)

\frac{1}{3}

JEE Advanced 2014

LEVELJEE Advanced

Let $x, y$ and $z$ be three vectors each of magnitude $2$ and the angle between each pair of them is $\frac{π}{3}$ . If $a$ is a non-zero vector perpendicular to $x$ and $y \times z$ and $b$ is a non-zero vector perpendicular to $y$ and $z \times x$ , then

* Multiple Correct Options

(A)

b = (b \cdot z) (z - x)

(B)

a = (a \cdot y) (y - z)

(C)

a \cdot b = - (a \cdot y) (b \cdot z)

(D)

a = - (a \cdot y) (z - y)

JEE Advanced 2004S

LEVELJEE Main

If $a = (\hat{i} + \hat{j} + \hat{k})$ , $a \cdot b = 1$ and $a \times b = \hat{j} - \hat{k}$ , then $b$ is

(A)

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

(B)

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

(C)

\hat{i}

(D)

2 \hat{i}

JEE Advanced 1995S

LEVELJEE Main

If $a, b, c$ are non coplanar unit vectors such that $a \times (b \times c) = \frac{( b + c )}{2}$ , then the angle between $a$ and $b$ is

(A)

3 π /4

(B)

π /4

(C)

π /2

(D)

π

JEE Advanced 2010

LEVELJEE Main

If $a$ and $b$ are vectors in space given by $a = \frac{i ^ - 2 j ^}{5}$ and $b = \frac{2 i ^ + j ^ + 3 k ^}{14}$ , then find the value of $(2 a + b) \cdot [(a \times b) \times (a - 2 b)]$ .

JEE Advanced 1992

LEVELJEE Advanced

A unit vector coplanar with $\hat{i} + \hat{j} + 2 \hat{k}$ and $\hat{i} + 2 \hat{j} + \hat{k}$ and perpendicular to $\hat{i} + \hat{j} + \hat{k}$ is .........

JEE Advanced 1999

LEVELJEE Advanced

Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

JEE Advanced 1997

LEVELJEE Main

If $A, B$ and $C$ are vectors such that $∣ B ∣ = ∣ C ∣$ . Prove that $[(A + B) \times (A + C)] \times (B \times C) \cdot (B + C) = 0$ .

JEE Advanced 1994

LEVELJEE Advanced

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

JEE Main 2017

LEVELJEE Main

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

(A)

\frac{25}{8}

(B)

2

(C)

5

(D)

\frac{1}{8}

JEE Main 2019 (9 January)

LEVELJEE Main

Let $a = \hat{i} - \hat{j}, b = \hat{i} + \hat{j} + \hat{k}$ and $c$ be a vector such that $a \times c + b = 0$ and $a \cdot c = 4$ , then $∣ c ∣^{2}$ is equal to :-

(A)

\frac{19}{2}

(B)

(C)

\frac{17}{2}

(D)

JEE Main 2019 (12 January)

LEVELJEE Main

Let $a, b$ and $c$ be three unit vectors, out of which vectors $b$ and $c$ are non-parallel. If $α$ and $β$ are the angles which vector $a$ makes with vectors $b$ and $c$ respectively and $a \times (b \times c) = \frac{1}{2} b$ , then $∣ α - β ∣$ is equal to :

(A)

6 0^{\circ}

(B)

3 0^{\circ}

(C)

9 0^{\circ}

(D)

4 5^{\circ}

JEE Main 2020 (9 January Shift 2)

LEVELJEE Main

Let $a, b$ and $c$ be three vectors such that $∣ a ∣ = 3, ∣ b ∣ = 5, b . c = 10$ and the angle between $b$ and $c$ is $\frac{π}{3}$ . If $a$ is perpendicular to vector $b \times c$ , then $∣ a \times (b \times c) ∣$ is equal to ________

JEE Main 2020 - 9 Jan (Evening)

LEVELJEE Main

Let $a$ , $b$ and $c$ be three vectors such that $∣ a ∣ = 3$ , $∣ b ∣ = 5$ , $b \cdot c = 10$ and angle between $b$ and $c$ is $\frac{π}{3}$ . If $a$ is perpendicular to the vector $b \times c$ , then $∣ a \times (b \times c) ∣$ is equal to

JEE Main 2020 - 8 Jan (Evening)

LEVELJEE Main

Let $a = \hat{i} - 2 \hat{j} + \hat{k}$ and $b = \hat{i} - \hat{j} + \hat{k}$ be two vectors. If $c$ is a vector such that $b \times c = b \times a$ and $c \cdot a = 0$ , then $c \cdot b$ is equal to

(A)

\frac{1}{2}

(B)

\frac{1}{3}

(C)

- \frac{1}{2}

(D)

- \frac{1}{3}

JEE Main 2020 - 4 Sep (Evening)

LEVELJEE Main

If $a = 2 \hat{i} + \hat{j} + 2 \hat{k}$ , then the value of $∣ \hat{i} \times (a \times \hat{i}) ∣^{2} + ∣ \hat{j} \times (a \times \hat{j}) ∣^{2} + ∣ \hat{k} \times (a \times \hat{k}) ∣^{2}$ is equal to

JEE Main 2021 (01 Sep Shift 2)

LEVELJEE Advanced

Let $a = 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $b = \hat{i} + 2 \hat{j} - \hat{k}$ . Let a vector $v$ be in the plane containing $a$ and $b$ . If $v$ is perpendicular to the vector $3 \hat{i} + 2 \hat{j} - \hat{k}$ and its projection on $a$ is 19 units, then $∣2 v ∣^{2}$ is equal to .

JEE Main 2021 (31 Aug Shift 2)

LEVELJEE Main

Let $a, b, c$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $r$ satisfies $a \times {(r - b) \times a} + b \times {(r - c) \times b} + c \times {(r - a) \times c} = 0$ , then $r$ is equal to :

(A)

\frac{1}{3} (a + b + c)

(B)

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

(C)

\frac{1}{2} (a + b + c)

(D)

\frac{1}{2} (a + b + 2 c)

JEE Main 2021 (26 Aug Shift 1)

LEVELJEE Advanced

Let $a = \hat{i} + \hat{j} + \hat{k}$ and $b = \hat{j} - \hat{k}$ . If $c$ is a vector such that $a \times c = b$ and $a \cdot c = 3$ , then $a \cdot (b \times c)$ is equal to :

JEE Main 2021 (20 July Shift 1)

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 $\frac{π}{6}$ , then the value of $∣ (a \times b) \times c ∣$ is :

(A)

\frac{2}{3}

(B)

(C)

(D)

\frac{3}{2}

JEE Main 2021 (27 July Shift 1)

LEVELJEE Main

Let $a = \hat{i} + \hat{j} + 2 \hat{k}$ and $b = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ . Then the vector product $(a + b) \times ((a \times ((a - b) \times b)) \times b)$ is equal to :

(A)

5 (34 \hat{i} - 5 \hat{j} + 3 \hat{k})

(B)

7 (34 \hat{i} - 5 \hat{j} + 3 \hat{k})

(C)

7 (30 \hat{i} - 5 \hat{j} + 7 \hat{k})

(D)

5 (30 \hat{i} - 5 \hat{j} + 7 \hat{k})

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b and c be three unit vectors such that a×(b×c)=23​​(b+c). If b is not parallel to c, then the angle between a and b is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a=10​1​(3i^+k^) and b=71​(2i^+3j^​−6k^), then the value of (2a−b)⋅[(a×b)×(a+2b)] is

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b and c be non-zero vectors such that (a×b)×c=31​∣b∣∣c∣a. If θ is the acute angle between the vectors b and c, then sinθ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a=(i^+j^​+k^), a⋅b=1 and a×b=j^​−k^, then b is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are non coplanar unit vectors such that a×(b×c)=2​(b+c)​, then the angle between a and b is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a and b are vectors in space given by a=5​i^−2j^​​ and b=14​2i^+j^​+3k^​, then find the value of (2a+b)⋅[(a×b)×(a−2b)].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A unit vector coplanar with i^+j^​+2k^ and i^+2j^​+k^ and perpendicular to i^+j^​+k^ is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let u and v be unit vectors. If w is a vector such that w+(w×u)=v, then prove that ∣(u×v)⋅w∣≤1/2 and that the equality holds if and only if u is perpendicular to v.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B and C are vectors such that ∣B∣=∣C∣. Prove that [(A+B)×(A+C)]×(B×C)⋅(B+C)=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors b,c,d are not coplanar, then prove that the vector (a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c) is parallel to a.

.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^​. Let c be a vector such that ∣c−a∣=3, ∣(a×b)×c∣=3 and the angle between c and a×b be 30∘. Then a⋅c is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=i^−j^​,b=i^+j^​+k^ and c be a vector such that a×c+b=0 and a⋅c=4, then ∣c∣2 is equal to :-

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

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=i^+j^​+2k^ and b=−i^+2j^​+3k^. Then the vector product (a+b)×((a×((a−b)×b))×b) is equal to :

Let $a, b$ and $c$ be three unit vectors such that $a \times (b \times c) = \frac{3}{2} (b + c)$ . If $b$ is not parallel to $c$ , then the angle between $a$ and $b$ is:

If $a = \frac{1}{10} (3 \hat{i} + \hat{k})$ and $b = \frac{1}{7} (2 \hat{i} + 3 \hat{j} - 6 \hat{k})$ , then the value of $(2 a - b) \cdot [(a \times b) \times (a + 2 b)]$ is

If $(a \times b) \times c = a \times (b \times c)$ where $a, b$ and $c$ are any three vectors such that $a \cdot b \neq = 0, b \cdot c \neq = 0$ then $a$ and $c$ are

Let $a, b$ and $c$ be non-zero vectors such that $(a \times b) \times c = \frac{1}{3} ∣ b ∣∣ c ∣ a$ . If $θ$ is the acute angle between the vectors $b$ and $c$ , then $sin θ$ equals

If $a = (\hat{i} + \hat{j} + \hat{k})$ , $a \cdot b = 1$ and $a \times b = \hat{j} - \hat{k}$ , then $b$ is

If $a, b, c$ are non coplanar unit vectors such that $a \times (b \times c) = \frac{( b + c )}{2}$ , then the angle between $a$ and $b$ is

If $a$ and $b$ are vectors in space given by $a = \frac{i ^ - 2 j ^}{5}$ and $b = \frac{2 i ^ + j ^ + 3 k ^}{14}$ , then find the value of $(2 a + b) \cdot [(a \times b) \times (a - 2 b)]$ .

A unit vector coplanar with $\hat{i} + \hat{j} + 2 \hat{k}$ and $\hat{i} + 2 \hat{j} + \hat{k}$ and perpendicular to $\hat{i} + \hat{j} + \hat{k}$ is .........

Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

If $A, B$ and $C$ are vectors such that $∣ B ∣ = ∣ C ∣$ . Prove that $[(A + B) \times (A + C)] \times (B \times C) \cdot (B + C) = 0$ .

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

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

Let $a = \hat{i} - \hat{j}, b = \hat{i} + \hat{j} + \hat{k}$ and $c$ be a vector such that $a \times c + b = 0$ and $a \cdot c = 4$ , then $∣ c ∣^{2}$ is equal to :-

Let $a = \hat{i} - 2 \hat{j} + \hat{k}$ and $b = \hat{i} - \hat{j} + \hat{k}$ be two vectors. If $c$ is a vector such that $b \times c = b \times a$ and $c \cdot a = 0$ , then $c \cdot b$ is equal to

If $a = 2 \hat{i} + \hat{j} + 2 \hat{k}$ , then the value of $∣ \hat{i} \times (a \times \hat{i}) ∣^{2} + ∣ \hat{j} \times (a \times \hat{j}) ∣^{2} + ∣ \hat{k} \times (a \times \hat{k}) ∣^{2}$ is equal to

Let $a = \hat{i} + \hat{j} + \hat{k}$ and $b = \hat{j} - \hat{k}$ . If $c$ is a vector such that $a \times c = b$ and $a \cdot c = 3$ , then $a \cdot (b \times c)$ is equal to :

Let $a = \hat{i} + \hat{j} + 2 \hat{k}$ and $b = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ . Then the vector product $(a + b) \times ((a \times ((a - b) \times b)) \times b)$ is equal to :