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Area of a parallelogram from its diagonals (or a side and a diagonal) in components

This type of question, asked 4 times: 3 May 2023 S1, 15 May 2023 S2, 22 Apr 2025 S2, 25 Apr 2025 S1. All 4 questions below, oldest first.

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  1. Question 1

    3 May 2023 S1

    112
    The area (in sq. units) of the parallelogram whose diagonals are along the vectors 8i^−6j^8\hat{i}-6\hat{j} and 3i^+4j^−12k^3\hat{i}+4\hat{j}-12\hat{k} is

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  2. Question 2

    15 May 2023 S2

    130
    One side and one diagonal of a parallelogram are represented by 3i^+j^−k^3\hat{i}+\hat{j}-\hat{k} and 2i^+j^−2k^2\hat{i}+\hat{j}-2\hat{k} respectively, then the area of parallelogram in square units is

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  3. Question 3

    22 Apr 2025 S2

    128
    If the area of parallelogram, whose diagonals are i^−j^+2k^\widehat{i}-\widehat{j}+ 2\widehat{k} and 2i^+3j^+αk^2\widehat{i}+ 3\widehat{j}+ \alpha\widehat{k} is 932\frac{\sqrt{93}}{2} sq. units, then α=\alpha =

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  4. Question 4

    25 Apr 2025 S1

    137
    If the area of a parallelogram whose diagonals are represented by vectors 3i^+λj^+2k^3\widehat{i}+\lambda\widehat{j}+ 2\widehat{k} and i^−2j^+3k^\widehat{i}- 2\widehat{j}+ 3\widehat{k} is 1172\frac{\sqrt{117}}{2} sq. units, then λ=\lambda=

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