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Vector c with conditions on a.c, |c - a| and the angle between c and a x b

This type of question, asked 10 times: 2 May 2023 S1, 3 May 2023 S1, 4 May 2023 S2, 9 May 2023 S1, 15 May 2023 S1, 10 May 2024 S1, 11 May 2024 S1, 11 May 2024 S2, 14 May 2024 S2, 15 Apr 2026 S2. All 10 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    104
    Let a⃗=2i^+j^−2k^\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} and b⃗=i^+j^\vec{b} = \hat{i} + \hat{j}. Let c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=3|\vec{c} - \vec{a}| = 3 and ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a} \times \vec{b}) \times \vec{c}| = 3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a} \times \vec{b} is 30∘30^\circ, then a⃗⋅c⃗\vec{a} \cdot \vec{c} is equal to

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  2. Question 2

    3 May 2023 S1

    141
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣, ∣c⃗−a⃗∣=22\vec{a}\cdot\vec{c}=|\vec{c}|,\ |\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is 30∘30^\circ, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is equal to

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  3. Question 3

    4 May 2023 S2

    122
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. Let c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=3|\vec{c}-\vec{a}|=3 and ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a}\times\vec{b})\times\vec{c}|=3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a}\times\vec{b} is 30°, then a⃗⋅c⃗\vec{a}\cdot\vec{c} is equal to

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  4. Question 4

    9 May 2023 S1

    133
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is 60∘60^\circ, then the value of ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is

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  5. Question 5

    15 May 2023 S1

    120
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is 30∘30^{\circ}, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is equal to

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  6. Question 6

    10 May 2024 S1

    119
    a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k}, b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. c⃗\vec{c} such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2}, angle between a⃗×b⃗\vec{a}\times\vec{b} and c⃗\vec{c} is π4\frac{\pi}{4}. ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is

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  7. Question 7

    11 May 2024 S1

    125
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is π6\frac{\pi}{6}, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is

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  8. Question 8

    11 May 2024 S2

    139
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k}, b⃗=i^+j^\vec{b}=\hat{i}+\hat{j} and c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=4|\vec{c}-\vec{a}|=4, ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a}\times\vec{b})\times\vec{c}|=3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a}\times\vec{b} is π6\frac{\pi}{6}, then a⃗⋅c⃗\vec{a}\cdot\vec{c} is equal to

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  9. Question 9

    14 May 2024 S2

    101
    Let a⃗=2i^+j^−2k^\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} and b⃗=i^+j^\vec{b} = \hat{i} + \hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣\vec{a} \cdot \vec{c} = |\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c} - \vec{a}| = 2\sqrt{2} and the angle between a⃗×b⃗\vec{a} \times \vec{b} and c⃗\vec{c} is 2π3\frac{2\pi}{3}, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a} \times \vec{b}) \times \vec{c}| =

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  10. Question 10

    15 Apr 2026 S2

    91
    Let a⃗=i^+2j^−2k^\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k} and b⃗=i^−j^+k^\vec{b} = \hat{i} - \hat{j} + \hat{k}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣,∣c⃗−a⃗∣=22\vec{a}\cdot\vec{c} = |\vec{c}|, |\vec{c} - \vec{a}| = 2\sqrt{2} and the angle between a⃗×b⃗\vec{a} \times \vec{b} and c⃗\vec{c} is 60∘60^\circ, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a} \times \vec{b}) \times \vec{c}| is equal to

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