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Vector r from r x b = c x b and r.a = 0

This type of question, asked 8 times: May 2022 S1, 2 May 2023 S2, 3 May 2023 S2, 14 May 2024 S2, 22 Apr 2025 S2, 20 Apr 2026 S1, 20 Apr 2026 S2, 15 May 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    May 2022 S1

    116
    Let a⃗=i^−2j^+k^\vec{a} = \hat{i} - 2\hat{j} + \hat{k} and b⃗=i^−j^+k^\vec{b} = \hat{i} - \hat{j} + \hat{k} be two vectors. If c⃗\vec{c} is a vector such that b⃗×c⃗=b⃗×a⃗\vec{b} \times \vec{c} = \vec{b} \times \vec{a} and c⃗⋅a⃗=0\vec{c} \cdot \vec{a} = 0, then c⃗⋅b⃗\vec{c} \cdot \vec{b} is equal to

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  2. Question 2

    2 May 2023 S2

    108
    Let A⃗=2i^+k^\vec{A} = 2\hat{i}+\hat{k}, B⃗=i^+j^+k^\vec{B} = \hat{i}+\hat{j}+\hat{k} and C⃗=4i^−3j^+7k^\vec{C} = 4\hat{i}-3\hat{j}+7\hat{k}. If a vector R⃗\vec{R} satisfies R⃗×B⃗=C⃗×B⃗\vec{R}\times\vec{B} = \vec{C}\times\vec{B} and R⃗⋅A⃗=0\vec{R}\cdot\vec{A} = 0, then R⃗\vec{R} is given by

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  3. Question 3

    3 May 2023 S2

    103
    The vectors a\mathbf{a} and b\mathbf{b} are not perpendicular and c\mathbf{c} and d\mathbf{d} are two vectors satisfying b×c=b×d\mathbf{b}\times\mathbf{c}=\mathbf{b}\times\mathbf{d} and a⋅d=0\mathbf{a}\cdot\mathbf{d}=0, then the vector d\mathbf{d} is equal to

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  4. Question 4

    14 May 2024 S2

    120
    Let a⃗=i^+2j^−k^\vec{a} = \hat{i}+2\hat{j}-\hat{k} and b⃗=i^+j^−k^\vec{b} = \hat{i}+\hat{j}-\hat{k}. If c⃗\vec{c} satisfies b⃗×c⃗=b⃗×a⃗\vec{b}\times\vec{c} = \vec{b}\times\vec{a} and c⃗⋅a⃗=0\vec{c}\cdot\vec{a} = 0, then c⃗⋅b⃗\vec{c}\cdot\vec{b} is

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  5. Question 5

    22 Apr 2025 S2

    125
    If pˉ=2i^+k^,qˉ=i^+j^+k^,rˉ=4i^−3j^+7k^\bar{p}= 2\widehat{i}+\widehat{k},\bar{q}=\widehat{i}+\widehat{j}+\widehat{k},\bar{r}= 4\widehat{i}- 3\widehat{j}+ 7\widehat{k} and a vector m‾\overline{m} is such that m‾×q‾=r‾×q‾,m‾⋅p‾=0\overline{m}\times\overline{q}=\overline{r}\times\overline{q},\overline{m}\cdot\overline{p}= 0, then m‾=\overline{m}=... .

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  6. Question 6

    20 Apr 2026 S1

    88
    Let aˉ=i^+j^+k^,bˉ=i^−3j^+2k^\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i} - 3\hat{j} + 2\hat{k} and cˉ=3i^−2k^\bar{c} = 3\hat{i} - 2\hat{k}. If a vector pˉ\bar{p} satisfies the conditions pˉ⋅cˉ=0\bar{p}\cdot\bar{c} = 0 and pˉ×aˉ=bˉ×aˉ\bar{p} \times \bar{a} = \bar{b} \times \bar{a}, then the value of ∣pˉ∣=|\bar{p}| =...

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  7. Question 7

    20 Apr 2026 S2

    88
    Let a⃗=2i^+k^,b⃗=i^+j^+k^\vec{a} = 2\hat{i} + \hat{k}, \vec{b} = \hat{i} + \hat{j} + \hat{k}, and c⃗=4i^−3j^+7k^\vec{c} = 4\hat{i} - 3\hat{j} + 7\hat{k}. If r⃗\vec{r} is a vector such that r⃗×b⃗=c⃗×b⃗\vec{r} \times \vec{b} = \vec{c} \times \vec{b} and r⃗⋅a⃗=0\vec{r}\cdot\vec{a} = 0, Then r⃗⋅c⃗=\vec{r}\cdot\vec{c} =

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  8. Question 8

    15 May 2026 S1

    88
    Let aˉ=2i^+k^\bar{a} = 2\hat{i} + \hat{k}, bˉ=i^+j^+k^\bar{b} = \hat{i} + \hat{j} + \hat{k}, and cˉ=4i^−3j^+7k^\bar{c} = 4\hat{i} - 3\hat{j} + 7\hat{k}. If rˉ\bar{r} is a vector such that rˉ×bˉ=cˉ×bˉ\bar{r} \times \bar{b} = \bar{c} \times \bar{b} and rˉ⋅aˉ=0\bar{r} \cdot \bar{a} = 0, then the value of the scalar product rˉ⋅i^\bar{r} \cdot \hat{i} is:

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