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Scalar triple product of combinations of a, b, c as a multiple of [a b c]

This type of question, asked 11 times: 2 May 2023 S1, 3 May 2023 S2, 9 May 2023 S1, 15 May 2023 S1, 16 May 2023 S1, 16 May 2023 S2, 9 May 2024 S1, 9 May 2024 S2, 26 Apr 2025 S1, 19 May 2026 S1, 19 May 2026 S2. All 11 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    112
    The scalar a⃗⋅[(b⃗+c⃗)×(a⃗+b⃗+c⃗)]\vec{a} \cdot [(\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})] equals

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  2. Question 2

    3 May 2023 S2

    110
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} are non-zero non-coplanar vectors and mm is non-zero scalar such that [ma⃗+b⃗    mb⃗+c⃗    mc⃗+a⃗]=28[a⃗ b⃗ c⃗][m\vec{a}+\vec{b}\;\; m\vec{b}+\vec{c}\;\; m\vec{c}+\vec{a}]=28[\vec{a}\,\vec{b}\,\vec{c}], then the value of mm is equal to

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  3. Question 3

    9 May 2023 S1

    121
    If a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} are three non-coplanar vectors, then (a⃗+b⃗+c⃗)⋅[(a⃗+b⃗)×(a⃗+c⃗)](\vec{a}+\vec{b}+\vec{c})\cdot[(\vec{a}+\vec{b})\times(\vec{a}+\vec{c})] equals

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  4. Question 4

    15 May 2023 S1

    146
    If u⃗,v⃗\vec{u}, \vec{v} and w⃗\vec{w} are three non-coplanar vectors, then (u⃗+v⃗−w⃗)⋅[(u⃗−v⃗)×(v⃗−w⃗)](\vec{u}+\vec{v}-\vec{w})\cdot[(\vec{u}-\vec{v})\times(\vec{v}-\vec{w})] is equal to

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  5. Question 5

    16 May 2023 S1

    105
    If a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are unit coplanar vectors, then the scalar triple product [2a⃗−b⃗,  2b⃗−c⃗,  2c⃗−a⃗][2\vec{a}-\vec{b},\; 2\vec{b}-\vec{c},\; 2\vec{c}-\vec{a}] has the value

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  6. Question 6

    16 May 2023 S2

    129
    If a⃗=a1i^+a2j^+a3k^\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}, b⃗=b1i^+b2j^+b3k^\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}, c⃗=c1i^+c2j^+c3k^\vec{c} = c_1\hat{i}+c_2\hat{j}+c_3\hat{k} and [3a⃗+b⃗,3b⃗+c⃗,3c⃗+a⃗]=λ[a⃗ b⃗ c⃗][3\vec{a}+\vec{b}, 3\vec{b}+\vec{c}, 3\vec{c}+\vec{a}] = \lambda[\vec{a}\,\vec{b}\,\vec{c}], then the value of λ\lambda is

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  7. Question 7

    9 May 2024 S1

    118
    If [(a⃗+2b⃗+3c⃗)×(b⃗+2c⃗+3a⃗)]⋅(c⃗+2a⃗+3b⃗)=54[(\vec{a}+2\vec{b}+3\vec{c})\times(\vec{b}+2\vec{c}+3\vec{a})]\cdot(\vec{c}+2\vec{a}+3\vec{b})=54, then the value of [a⃗ b⃗ c⃗][\vec{a}\,\vec{b}\,\vec{c}] is

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  8. Question 8

    9 May 2024 S2

    122
    If a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} are any three non-zero vectors, then (a⃗+2b⃗+c⃗)⋅[(a⃗−b⃗)×(a⃗−b⃗−c⃗)]=(\vec{a}+2\vec{b}+\vec{c})\cdot[(\vec{a}-\vec{b})\times(\vec{a}-\vec{b}-\vec{c})]=

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  9. Question 9

    26 Apr 2025 S1

    150
    The volume of tetrahedron with co-terminus edges aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} is 643\frac{64}{3} cubic units, then volume of parallelopiped considering co-terminus edges given by the vectors aˉ+bˉ,bˉ+cˉ,cˉ+aˉ\bar{a}+\bar{b},\bar{b}+\bar{c},\bar{c}+\bar{a} is ____\_\_\_\_ cubic units.

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  10. Question 10

    19 May 2026 S1

    90
    The scalar triple product of the vectors [2aˉ+3bˉ  3bˉ+cˉ  cˉ+2aˉ][2\bar{a} + 3\bar{b}\ \ 3\bar{b} + \bar{c}\ \ \bar{c} + 2\bar{a}] is equal to:

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  11. Question 11

    19 May 2026 S2

    92
    If three coterminous edges of a parallelopiped are represented by a⃗−b⃗,b⃗−c⃗\vec{a} - \vec{b}, \vec{b} - \vec{c} and c⃗−a⃗\vec{c} - \vec{a} then its volume is

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