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Each of a, b, c perpendicular to the sum of the other two, magnitudes given: find |a + b + c|

The same question, asked 8 times: 9 May 2023 S2*, 10 May 2023 S1*, 11 May 2024 S1, 22 Apr 2025 S2*, 26 Apr 2025 S1*, 19 Apr 2026 S1*, 19 Apr 2026 S2*, 12 May 2026 S2. Every wording below, oldest first.

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  1. Question 1

    9 May 2023 S2

    108
    Let a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} be vectors of magnitude 2, 3 and 4 respectively. If a⃗\vec{a} is perpendicular to (b⃗+c⃗)(\vec{b}+\vec{c}), b⃗\vec{b} is perpendicular to (c⃗+a⃗)(\vec{c}+\vec{a}) and c⃗\vec{c} is perpendicular to (a⃗+b⃗)(\vec{a}+\vec{b}), then the magnitude of a⃗+b⃗+c⃗\vec{a}+\vec{b}+\vec{c} is equal to

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  2. Question 2

    10 May 2023 S1

    139
    Let A⃗,B⃗,C⃗\vec{A},\vec{B},\vec{C} be vectors of lengths 3 units, 4 units, 5 units respectively. Let A⃗⊥(B⃗+C⃗)\vec{A}\perp(\vec{B}+\vec{C}), B⃗⊥(C⃗+A⃗)\vec{B}\perp(\vec{C}+\vec{A}) and C⃗⊥(A⃗+B⃗)\vec{C}\perp(\vec{A}+\vec{B}), then the length of vector A⃗+B⃗+C⃗\vec{A}+\vec{B}+\vec{C} is

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  3. Question 3

    11 May 2024 S1

    124
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three vectors such that a⃗⋅(b⃗+c⃗)+b⃗⋅(c⃗+a⃗)+c⃗⋅(a⃗+b⃗)=0\vec{a}\cdot(\vec{b}+\vec{c}) + \vec{b}\cdot(\vec{c}+\vec{a}) + \vec{c}\cdot(\vec{a}+\vec{b}) = 0 and ∣a⃗∣=1|\vec{a}|=1, ∣b⃗∣=8|\vec{b}|=8 and ∣c⃗∣=4|\vec{c}|=4, then ∣a⃗+b⃗+c⃗∣|\vec{a}+\vec{b}+\vec{c}| has the value

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  4. Question 4

    22 Apr 2025 S2

    139
    If the lengths of three vectors aˉ,bˉ\bar{a},\bar{b} and cˉ\bar{c} are 5,12,135,12,13 units respectively, and each one is perpendicular to the sum of the other two, then ∣a‾+b‾+c‾∣=|\overline{a}+\overline{b}+\overline{c}| =

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  5. Question 5

    26 Apr 2025 S1

    123
    Let aˉ,bˉ\bar{a},\bar{b} and cˉ\bar{c} be vectors of magnitude 2,3 and 4 respectively. If aˉ\bar{a} is perpendicular to (b‾+c‾\overline{b}+\overline{c}), b‾\overline{b} is perpendicular to (c‾+a‾\overline{c}+\overline{a}) and c‾\overline{c} is perpendicular to (aˉ+bˉ)(\bar{a}+\bar{b}), then the magnitude of a‾+b‾+c‾\overline{a}+\overline{b}+\overline{c} is

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  6. Question 6

    19 Apr 2026 S1

    89
    If ∣a⃗∣=3,∣b⃗∣=4,∣c⃗∣=5|\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5 such that each vector is perpendicular to the sum of the other two, then ∣a⃗+b⃗+c⃗∣|\vec{a} + \vec{b} + \vec{c}| is equal to

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  7. Question 7

    19 Apr 2026 S2

    87
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three vectors such that a⃗⊥(b⃗+c⃗),b⃗⊥(c⃗+a⃗)\vec{a} \perp (\vec{b} + \vec{c}), \vec{b} \perp (\vec{c} + \vec{a}), and c⃗⊥(a⃗+b⃗)\vec{c} \perp (\vec{a} + \vec{b}) and ∣a⃗∣=1,∣b⃗∣=2,∣c⃗∣=3|\vec{a}| = 1, |\vec{b}| = 2, |\vec{c}| = 3, then ∣a⃗+b⃗+c⃗∣|\vec{a} + \vec{b} + \vec{c}| is...

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  8. Question 8

    12 May 2026 S2

    89
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three vectors such that a⃗⋅(b⃗+c⃗)+b⃗⋅(c⃗+a⃗)+c⃗⋅(a⃗+b⃗)=0\vec{a}\cdot(\vec{b} + \vec{c}) + \vec{b}\cdot(\vec{c} + \vec{a}) + \vec{c}\cdot(\vec{a} + \vec{b}) = 0 and ∣a⃗∣=8,∣b⃗∣=1,∣c⃗∣=4|\vec{a}| = 8, |\vec{b}| = 1, |\vec{c}| = 4, then ∣a⃗+b⃗+c⃗∣=…|\vec{a} + \vec{b} + \vec{c}| = \ldots

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