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Vector perpendicular to two given vectors (a multiple of their cross product) fixed by one more condition

This type of question, asked 4 times: 2021 Paper 6, 2021 Paper 14, 10 Apr 2023, 29 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 6

    89
    Let c→\overrightarrow{c} be a vector perpendicular to the vectors a→=i^+j^−k^\overrightarrow{a}=\widehat{i}+\widehat{j}-\widehat{k} and b→=i^+2j^+k^\overrightarrow{b}=\widehat{i}+ 2\widehat{j}+\widehat{k}. If c→⋅(i^+j^+3k^)=8\overrightarrow{c}\cdot (\widehat{i}+\widehat{j}+ 3\widehat{k}) = 8 then the value of c→⋅(a→×b→)\overrightarrow{c}\cdot (\overrightarrow{a}\times\overrightarrow{b}) is equal to

  2. Question 2

    2021 Paper 14

    60
    p→=(3i^+2j^+k^),q→=(2i^+j^+k^),r→\overrightarrow{p}= (3\widehat{i}+ 2\widehat{j}+\widehat{k}),\overrightarrow{q}= (2\widehat{i}+\widehat{j}+\widehat{k}),\overrightarrow{r} is perpendicular to both p→+q→\overrightarrow{p}+\overrightarrow{q} and (p→−q→),∣r→∣=3,r→=ai^+bj^+ck^(\overrightarrow{p}-\overrightarrow{q}),|\overrightarrow{r}| =\sqrt{3},\overrightarrow{r}=a\widehat{i}+b\widehat{j}+c\widehat{k}. Find ∣a∣+∣b∣+∣c∣|a| + |b| + |c|.

  3. Question 3

    10 Apr 2023

    161
    Let a→=2i^+7j^−k^,b→=3i^+5k^\overrightarrow{a}= 2\widehat{i}+ 7\widehat{j}-\widehat{k},\overrightarrow{b}= 3\widehat{i}+ 5\widehat{k} and c→=i^−j^+2k^\overrightarrow{c}=\widehat{i}-\widehat{j}+ 2\widehat{k} . Let d→\overrightarrow{d} be a vector which is perpendicular to both a→\overrightarrow{a} and  b→,\ \overrightarrow{b}, and  c→⋅d→=12\ \overrightarrow{c}\cdot\overrightarrow{d}= 12. Then (−i^+j^−k^)⋅(c→×d→)( -\widehat{i}+\widehat{j}-\widehat{k}) \cdot (\overrightarrow{c}\times\overrightarrow{d}) is equal to

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  4. Question 4

    10 Apr 2023

    61
    Let OO be the origin and the position vector of the point PP be −i^−2j^+3k^-\widehat{i}- 2\widehat{j}+ 3\widehat{k}. If the position vectors of the points A,BA,B and CC are −2i^+j^−3k^,2i^+4j^−2k^- 2\widehat{i}+\widehat{j}- 3\widehat{k},2\widehat{i}+ 4\widehat{j}- 2\widehat{k} and −4i^+2j^−k^- 4\widehat{i}+ 2\widehat{j}-\widehat{k} respectively then the projection of the vector OP→\overrightarrow{OP} on a vector perpendicular to the vectors AB→\overrightarrow{AB} and AC→\overrightarrow{AC} is

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  5. Question 5

    29 Jan 2025

    144
    Let â be a unit vector perpendicular to the vectors b→=i^−2j^+3k^\overrightarrow{b}=\widehat{i}- 2\widehat{j}+ 3\widehat{k} and c→=2i^+3j^−k^\overrightarrow{c}= 2\widehat{i}+ 3\widehat{j}-\widehat{k}, and makes an angle of cos⁡−1(−13)\cos^{- 1}\left( -\frac{1}{3} \right) with the vector i^+j^+k^\widehat{i}+\widehat{j}+\widehat{k}. If a^\widehat{a} makes an angle of π3\frac{\pi}{3} with the vector i^+αj^+k^\widehat{i}+ \alpha\widehat{j}+\widehat{k}, then the value of α\alpha is :

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