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Vector in the plane of a and b (written as la + mb) fixed by two conditions

This type of question, asked 8 times: 2021 Paper 1, 2021 Paper 8, 2021 Paper 24, 26 Jun 2022, 28 Jun 2022, 24 Jan 2025, 24 Jan 2026 S2, 2 Apr 2026 S2. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    87
    Let three vectors a→,b→\overrightarrow{a},\overrightarrow{b} and c→\overrightarrow{c} be such that c→\overrightarrow{c} is coplanar with a→\overrightarrow{a} and b→,a→.c→=7\overrightarrow{b},\overrightarrow{a}.\overrightarrow{c}= 7 and b→\overrightarrow{b} is perpendicular to c→\overrightarrow{c}, where a→=−i^+j^+k^\overrightarrow{a}= -\widehat{i}+\widehat{j}+\widehat{k} and b→=2i^+k^\overrightarrow{b}= 2\widehat{i}+\widehat{k}, then the value of 2∣a→+b→+c→∣22\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|^{2} is

  2. Question 2

    2021 Paper 8

    88
    Let x→\overrightarrow{x} be a vector in the plane containing vectors a→=2i^−j^+k^\overrightarrow{a}= 2\widehat{i}-\widehat{j}+\widehat{k} and b→=i^+2j^−k^\overrightarrow{b}=\widehat{i}+ 2\widehat{j}-\widehat{k}. If the vector x→\overrightarrow{x} is perpendicular to (3i^+2j^−k^)(3\widehat{i}+ 2\widehat{j}-\widehat{k}) and its projection on a→\overrightarrow{a} is 1762\frac{17\sqrt{6}}{2}, then the value of ∣x→∣2|\overrightarrow{x}|^{2} is equal to

  3. Question 3

    2021 Paper 24

    88
    Let a→=2i^−j^+2k^\overrightarrow{a}= 2\widehat{i}-\widehat{j}+ 2\widehat{k} and b→=i^+2j^−k^\overrightarrow{b}=\widehat{i}+ 2\widehat{j}-\widehat{k}. Let a vector v→\overrightarrow{v} be in the plane containing a→\overrightarrow{a} and b→\overrightarrow{b}. If v→\overrightarrow{v} is perpendicular to the vector 3i^+2j^−k^3\widehat{i}+ 2\widehat{j}-\widehat{k} and its projection on a→\overrightarrow{a} is 19 units, then ∣2v→∣2|2\overrightarrow{v}|^{2} is equal to

  4. Question 4

    26 Jun 2022

    166
    Let a→=i^+j^+2k^,b→=2i^−3j^+k^\overrightarrow{a}=\widehat{i}+\widehat{j}+ 2\widehat{k},\overrightarrow{b}= 2\widehat{i}- 3\widehat{j}+\widehat{k} and c→=i^−j^+k^\overrightarrow{c}=\widehat{i}-\widehat{j}+\widehat{k} be three given vectors. Let v→\overrightarrow{v} be a vector in the plane of a→\overrightarrow{a} and b→\overrightarrow{b} whose projection on c→\overrightarrow{c} is 23\frac{2}{\sqrt{3}}. If v→⋅j^=7\overrightarrow{v}\cdot\widehat{j}= 7, then v→⋅(i^+k^)\overrightarrow{v}\cdot (\widehat{i}+\widehat{k}) is equal to:

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  5. Question 5

    28 Jun 2022

    84
    If a→=2i^+j^+3k^, b→=3i^+3j^+k^\overrightarrow{a}= 2\widehat{i}+\widehat{j}+ 3\widehat{k},\ \overrightarrow{b}= 3\widehat{i}+ 3\widehat{j}+\widehat{k} and c→=c1i^+c2j^+c3k^\overrightarrow{c}=c_{1}\widehat{i}+c_{2}\widehat{j}+c_{3}\widehat{k} are coplanar vectors and a→⋅c→=5,b→⊥c→\overrightarrow{a}\cdot\overrightarrow{c}= 5,\overrightarrow{b}\bot\overrightarrow{c}, then 122(c1+c2+c3)122\left( c_{1}+c_{2}+c_{3} \right) is equal to

  6. Question 6

    24 Jan 2025

    51
    Let a→=i^+2j^+3k^,b→=3i^+j^−k^\overrightarrow{a}=\widehat{i}+ 2\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 3\widehat{i}+\widehat{j}-\widehat{k} and c→\overrightarrow{c} be three vectors such that c→\overrightarrow{c} is coplanar with a→\overrightarrow{a} and b→\overrightarrow{b}. If the vector c→\overrightarrow{c} is perpendicular to b→\overrightarrow{b} and a→.c→=5\overrightarrow{a}.\overrightarrow{c}= 5, then ∣c→∣|\overrightarrow{c}| is equal to

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  7. Question 7

    24 Jan 2026 S2

    54
    Let  a→=2i^−j^−k^,b→=i^+3j^−k^\ \overrightarrow{a}= 2\widehat{i}-\widehat{j}-\widehat{k},\overrightarrow{b}=\widehat{i}+ 3\widehat{j}-\widehat{k} and c→=2i^+j^+3k^\overrightarrow{c}= 2\widehat{i}+\widehat{j}+ 3\widehat{k}. Let v→\overrightarrow{v} be the vector in the plane of the vectors a→\overrightarrow{a} and b→\overrightarrow{b}, such that the length of its projection on the vector c→\overrightarrow{c} is 114\frac{1}{\sqrt{14}}. Then ∣v→∣|\overrightarrow{v}| is equal to

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  8. Question 8

    2 Apr 2026 S2

    64
    Let the vectors a→=−i^+j^+3k^\overrightarrow{a}= -\widehat{i}+\widehat{j}+ 3\widehat{k} and b→=i^+3j^+k^\overrightarrow{b}=\widehat{i}+ 3\widehat{j}+\widehat{k}. For some λ,μ∈R\lambda,\mu\in R, let c→=λa→+μb→\overrightarrow{c}=\lambda\overrightarrow{a}+\mu\overrightarrow{b}. If c→⋅(3i^−6j^+2k^)=10\overrightarrow{c}\cdot (3\widehat{i}- 6\widehat{j}+ 2\widehat{k}) = 10 and c→⋅(i^+j^+k^)=−2\overrightarrow{c}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) = - 2, then ∣c→∣2|\overrightarrow{c}|^{2} is equal to :

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