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Altitude of a parallelepiped on a given base

This type of question, asked 3 times: 11 Apr 2026 S2, 15 Apr 2026 S2, 19 May 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    11 Apr 2026 S2

    91
    The parallelopiped is determined by vectors aˉ=−2i^+5j^+3k^,bˉ=i^+3j^−2k^,cˉ=−3i^+j^+4k^\bar{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}, \bar{b} = \hat{i} + 3\hat{j} - 2\hat{k}, \bar{c} = -3\hat{i} + \hat{j} + 4\hat{k}. The altitude of parallelopiped on the parallelogram base determined by vectors bˉ\bar{b} and cˉ\bar{c} is

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  2. Question 2

    15 Apr 2026 S2

    89
    The altitude of the parallelopiped, whose coterminous edges are the vectors a⃗=i^+j^+k^\vec{a} = \hat{i} + \hat{j} + \hat{k}, b⃗=2i^+4j^−k^\vec{b} = 2\hat{i} + 4\hat{j} - \hat{k}, c⃗=i^+j^+3k^\vec{c} = \hat{i} + \hat{j} + 3\hat{k}, where a⃗,b⃗\vec{a}, \vec{b} are the sides of the base of parallelopiped, is

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  3. Question 3

    19 May 2026 S2

    91
    A parallelopiped is determined by vectors a⃗=i^+2k^,b⃗=i^+j^−k^\vec{a} = \hat{i} + 2\hat{k}, \vec{b} = \hat{i} + \hat{j} - \hat{k} and c⃗=2i^+2j^+3k^\vec{c} = 2\hat{i} + 2\hat{j} + 3\hat{k}. The length (in units) of its altitude drawn to the base determined by vectors b⃗\vec{b} and c⃗\vec{c} is

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