Scalar Triple Product

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0

1

2

3

(p, q)

(p, q)

(p, q)

(p, q)

- 4

- 2

a

b

a

b

a

b

λ

λ

λ

λ

y

x

x

y

x

y

λ

λ

λ

λ

3 u \cdot v \times w

u \cdot v \times w

u \cdot w \times v

\frac{1}{2}

\frac{1}{2 2}

\frac{3}{2}

\frac{1}{3}

- 3

3

1/ 3

3

- 1

10 + 6

59

60

x

y

x

y

x

y

0

1

- 3

3

α = 1, β = - 1

α = 1, β = \pm 1

α = - 1, β = \pm 1

α = \pm 1, β = 1

u \cdot (v \times w)

(v \times w) \cdot u

v \cdot (u \times w)

(u \times v) \cdot w

u \cdot (v \times w)

(u \cdot v) \cdot w

(u \cdot v) w

u \times (v \cdot w)

\pm \frac{i ^ + j ^ - 2 k ^}{6}

\pm \frac{i ^ + j ^ - k ^}{3}

\pm \frac{i ^ + j ^ + k ^}{3}

\pm \hat{k}

0

[a b c]

2 [a b c]

- [a b c]

a

b

a

b

a

b

\frac{1}{4} (a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) (b_{1}^{2} + b_{2}^{2} + b_{3}^{2})

\frac{3}{4} (a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) (b_{1}^{2} + b_{2}^{2} + b_{3}^{2}) (c_{1}^{2} + c_{2}^{2} + c_{3}^{2})

4/13

2/7

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

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

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

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

.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]=λ[a b c]2 then λ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If u,v,w are non-coplanar vectors and p,q are real numbers, then the equality [3u pv pw]−[pv w qu]−[2w qv qu]=0 holds for :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=i^+j^​+k^,b=i^−j^​+2k^ and c=xi^+(x−2)j^​−k^. If the vectors c lies in the plane of a and b, then x equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b and c be distinct non-negative numbers. If the vectors ai^+aj^​+ck^, i^+k^ and ci^+cj^​+bk^ lie in a plane, then c 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 vectors and λ is a real number then [λ(a+b)λ2bλc]=[ab+cb] for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=i^−k^,b=xi^+j^​+(1−x)k^ and c=yi^+xj^​+(1+x−y)k^. Then [abc] depends on

.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 vectors and λ is a real number, then the vectors a+2b+3c, λb+4c and (2λ−1)c are non coplanar for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If u,v and w are three non-coplanar vectors, then (u+v−w)⋅[(u−v)×(v−w)] equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are vectors such that [a b c]=4 then [a×b b×c c×a]=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of distinct real values of λ, for which the vectors −λ2i^+j^​+k^, i^−λ2j^​+k^ and i^+j^​−λ2k^ are coplanar, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 'a' so that the volume of parallelopiped formed by i^+aj^​+k^, j^​+ak^ and ai^+k^ becomes minimum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let V=2i^+j^​−k^ and W=i^+3k^. If U is a unit vector, then the maximum value of the scalar triple product [UVW] is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=i^−k^,b=xi^+j^​+(1−x)k^ and c=yi^+xj^​+(1+x−y)k^. Then [abc] depends on

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b and c are unit coplanar vectors, then the scalar triple product [2a−b,2b−c,2c−a]=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a=i^+j^​+k^, b=4i^+3j^​+4k^ and c=i^+αj^​+βk^ are linearly dependent vectors and ∣c∣=3​, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For three vectors u,v,w which of the following expression is not equal to any of the remaining three?

Which of the following expressions are meaningful?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=i^−j^​, b=j^​−k^, c=k^−i^. If d is a unit vector such that a⋅d=0=[bcd], then d equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b and c are three non coplanar vectors, then (a+b+c)⋅[(a+b)×(a+c)] equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be distinct non-negative numbers. If the vectors ai^+aj^​+ck^,i^+k^ and ci^+cj^​+bk^ lie in a plane, then c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The volume of the parallelopiped whose sides are given by OA=2i^−2j^​,OB=i^+j^​−k^,OC=3i^−k^, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For non-zero vectors a,b,c, ∣(a×b)⋅c∣=∣a∣∣b∣∣c∣ holds if and only if

If $[a \times b b \times c c \times a] = λ [a b c]^{2}$ then $λ$ is equal to

If $u, v, w$ are non-coplanar vectors and $p, q$ are real numbers, then the equality $[3 u p v p w] - [p v w q u] - [2 w q v q u] = 0$ holds for :

Let $a = \hat{i} + \hat{j} + \hat{k}, b = \hat{i} - \hat{j} + 2 \hat{k}$ and $c = x \hat{i} + (x - 2) \hat{j} - \hat{k}$ . If the vectors $c$ lies in the plane of $a$ and $b$ , then $x$ equals

Let $a, b$ and $c$ be distinct non-negative numbers. If the vectors $a \hat{i} + a \hat{j} + c \hat{k}$ , $\hat{i} + \hat{k}$ and $c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane, then $c$ is

If $a, b, c$ are non coplanar vectors and $λ$ is a real number then $[λ (a + b) λ^{2} b λ c] = [a b + c b]$ for

Let $a = \hat{i} - \hat{k}, b = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $c = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ . Then $[a b c]$ depends on

If $a, b, c$ are non-coplanar vectors and $λ$ is a real number, then the vectors $a + 2 b + 3 c$ , $λ b + 4 c$ and $(2 λ - 1) c$ are non coplanar for

If $u, v$ and $w$ are three non-coplanar vectors, then $(u + v - w) \cdot [(u - v) \times (v - w)]$ equals

If $a, b, c$ are vectors such that $[a b c] = 4$ then $[a \times b b \times c c \times a] =$

The number of distinct real values of $λ$ , for which the vectors $- λ^{2} \hat{i} + \hat{j} + \hat{k}$ , $\hat{i} - λ^{2} \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - λ^{2} \hat{k}$ are coplanar, is

The value of 'a' so that the volume of parallelopiped formed by $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

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

Let $a = \hat{i} - \hat{k}, b = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $c = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ . Then $[a b c]$ depends on

If $a, b$ and $c$ are unit coplanar vectors, then the scalar triple product $[2 a - b, 2 b - c, 2 c - a] =$

If $a = \hat{i} + \hat{j} + \hat{k}$ , $b = 4 \hat{i} + 3 \hat{j} + 4 \hat{k}$ and $c = \hat{i} + α \hat{j} + β \hat{k}$ are linearly dependent vectors and $∣ c ∣ = 3$ , then

For three vectors $u, v, w$ which of the following expression is not equal to any of the remaining three?

Let $a = \hat{i} - \hat{j}$ , $b = \hat{j} - \hat{k}$ , $c = \hat{k} - \hat{i}$ . If $d$ is a unit vector such that $a \cdot d = 0 = [b c d]$ , then $d$ equals

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

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}$ and $c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane, then $c$ is

The volume of the parallelopiped whose sides are given by $O A = 2 \hat{i} - 2 \hat{j}, O B = \hat{i} + \hat{j} - \hat{k}, O C = 3 \hat{i} - \hat{k}$ , is

For non-zero vectors $a, b, c$ , $∣ (a \times b) \cdot c ∣ = ∣ a ∣∣ b ∣∣ c ∣$ holds if and only if