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Projections of v and w on u equal and v perpendicular to w: find |u - v + w|

The same question, asked 7 times: 3 May 2023 S1*, 16 May 2023 S1*, 12 May 2024 S1, 14 May 2024 S2, 19 Apr 2025 S2, 26 Apr 2025 S2*, 18 Apr 2026 S1. Every wording below, oldest first.

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  1. Question 1

    3 May 2023 S1

    139
    Let the vectors a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be such that ∣a⃗∣=2,∣b⃗∣=4,∣c⃗∣=4|\vec{a}|=2,|\vec{b}|=4,|\vec{c}|=4. If the projection of b⃗\vec{b} on a⃗\vec{a} equals the projection of c⃗\vec{c} on a⃗\vec{a} and b⃗\vec{b} is perpendicular to c⃗\vec{c}, then the value of ∣a⃗+b⃗−c⃗∣|\vec{a}+\vec{b}-\vec{c}| is equal to

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  2. Question 2

    16 May 2023 S1

    107
    Let the vectors a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be such that ∣a⃗∣=2,∣b⃗∣=4|\vec{a}|=2, |\vec{b}|=4 and ∣c⃗∣=4|\vec{c}|=4. If the projection of b⃗\vec{b} on a⃗\vec{a} is equal to the projection of c⃗\vec{c} on a⃗\vec{a} and b⃗\vec{b} is perpendicular to c⃗\vec{c}, then the value of ∣a⃗+b⃗−c⃗∣|\vec{a}+\vec{b}-\vec{c}| is

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  3. Question 3

    12 May 2024 S1

    135
    u⃗\vec{u}, v⃗\vec{v}, w⃗\vec{w} are three vectors such that ∣u⃗∣=1|\vec{u}|=1, ∣v⃗∣=2|\vec{v}|=2, ∣w⃗∣=3|\vec{w}|=3. If the projection of v⃗\vec{v} along u⃗\vec{u} is equal to projection of w⃗\vec{w} along u⃗\vec{u} and v⃗\vec{v}, w⃗\vec{w} are perpendicular to each other, then ∣u⃗−v⃗+w⃗∣=|\vec{u}-\vec{v}+\vec{w}|=

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  4. Question 4

    14 May 2024 S2

    110
    Let u⃗,v⃗,w⃗\vec{u}, \vec{v}, \vec{w} be vectors with ∣u⃗∣=1,∣v⃗∣=2,∣w⃗∣=3|\vec{u}|=1, |\vec{v}|=2, |\vec{w}|=3. Projection of v⃗\vec{v} along u⃗\vec{u} equals projection of w⃗\vec{w} along u⃗\vec{u}, and v⃗⊥w⃗\vec{v} \perp \vec{w}. Then ∣u⃗−v⃗+w⃗∣|\vec{u} - \vec{v} + \vec{w}| equals

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  5. Question 5

    19 Apr 2025 S2

    110
    Let u‾,v‾,w‾\overline{u},\overline{v},\overline{w} be the vectors such that ∣u‾∣=1,∣v‾∣=2,∣w‾∣=3.|\overline{u}| = 1,|\overline{v}| = 2,|\overline{w}| = 3. If the projection v‾\overline{v} along u‾\overline{u} is equal to that of w‾\overline{w} along u‾\overline{u} and the vectors vˉ,wˉ\bar{v},\bar{w} are perpendicular to each other then ∣u‾−v‾+w‾∣|\overline{u}-\overline{v}+\overline{w}| equals

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  6. Question 6

    26 Apr 2025 S2

    129
    The vectors aˉ,bˉ\bar{a},\bar{b} and cˉ\bar{c} are such that ∣aˉ∣=2,∣bˉ∣=4,∣cˉ∣=4|\bar{a}| = 2,|\bar{b}| = 4,|\bar{c}| = 4. If the projection of bˉ\bar{b} on aˉ\bar{a} is equal to projection of cˉ\bar{c} on aˉ\bar{a} and bˉ\bar{b} is perpendicular to cˉ\bar{c}, then the value of ∣aˉ+bˉ−cˉ∣|\bar{a}+\bar{b}-\bar{c}| is

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  7. Question 7

    18 Apr 2026 S1

    89
    Let uˉ,vˉ,wˉ\bar{u}, \bar{v}, \bar{w} be three vectors such that ∣uˉ∣=1,∣vˉ∣=2,∣wˉ∣=3|\bar{u}| = 1, |\bar{v}| = 2, |\bar{w}| = 3. If the projection of vˉ\bar{v} along uˉ\bar{u} is equal to the projection of wˉ\bar{w} along uˉ\bar{u} and vˉ,wˉ\bar{v}, \bar{w} are perpendicular to each other, then ∣uˉ−vˉ+wˉ∣=|\bar{u} - \bar{v} + \bar{w}| =...

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