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a perpendicular to both b and c: [a b c] or a in terms of b x c

This type of question, asked 3 times: 16 May 2023 S2, 21 Apr 2025 S1, 17 Apr 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    16 May 2023 S2

    117
    If a⃗\vec{a} is perpendicular to b⃗\vec{b} and c⃗\vec{c}, ∣a⃗∣=2|\vec{a}|=2, ∣b⃗∣=3|\vec{b}|=3, ∣c⃗∣=4|\vec{c}|=4 and the angle between b⃗\vec{b} and c⃗\vec{c} is π3\frac{\pi}{3}, then [a⃗ b⃗ c⃗]=[\vec{a}\,\vec{b}\,\vec{c}] =

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  2. Question 2

    21 Apr 2025 S1

    114
    Let a‾,b‾\overline{a},\overline{b}, and c‾\overline{c} be unit vectors. Suppose that aˉ⋅bˉ=aˉ⋅cˉ=0\bar{a}\cdot\bar{b}=\bar{a}\cdot\bar{c}= 0 and if the angle between bˉ\bar{b} and cˉ\bar{c} is π6\frac{\pi}{6}, then aˉ\bar{a} is

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  3. Question 3

    17 Apr 2026 S1

    91
    Let aˉ=(a1i^+a2j^+a3k^), bˉ=(b1i^+b2j^+b3k^), cˉ=(c1i^+c2j^+c3k^)\bar{a} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}),\ \bar{b} = (b_1\hat{i} + b_2\hat{j} + b_3\hat{k}),\ \bar{c} = (c_1\hat{i} + c_2\hat{j} + c_3\hat{k}) be three non-zero vectors such that aˉ\bar{a} is a unit vector perpendicular to both bˉ\bar{b} and cˉ\bar{c}. If the angle between bˉ\bar{b} and cˉ\bar{c} is π3\dfrac{\pi}{3} then ∣a1a2a3b1b2b3c1c2c3∣2=\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}^2 =

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