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Length of the angle bisector from a vertex of a triangle with given position vectors

The same question, asked 2 times: 27 Jan 2024*, 29 Jan 2024. Every wording below, oldest first.

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  1. Question 1

    27 Jan 2024

    166
    The position vectors of the vertices A,BA,B and CC of a triangle are 2i^−3j^+3k^,2i^+2j^+3k^2\widehat{i}- 3\widehat{j}+ 3\widehat{k},2\widehat{i}+ 2\widehat{j}+ 3\widehat{k} and −i^+j^+3k^-\widehat{i}+\widehat{j}+ 3\widehat{k} respectively. Let ll denotes the length of the angle bisector ADAD of ∠BAC\angle BAC where DD is on the line segment BCBC, then 2l22l^{2} equals:

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  2. Question 2

    29 Jan 2024

    73
    Let OO be the origin and the position vector of AA and BB be 2i^+2j^+k^2\widehat{i}+ 2\widehat{j}+\widehat{k} and 2i^+4j^+4k^2\widehat{i}+ 4\widehat{j}+ 4\widehat{k} respectively. If the internal bisector of ∠AOB\angle AOB meets the line ABAB at CC, then the length of OCOC is

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