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Dot products and angles of vectors built from a, b and a x b (a x b is perpendicular to both)

This type of question, asked 8 times: 2021 Paper 10, 27 Jun 2022, 29 Jul 2022, 30 Jan 2023, 30 Jan 2024, 31 Jan 2024, 7 Apr 2025, 4 Apr 2026 S2. All 9 questions below, oldest first.

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  1. Question 1

    2021 Paper 10

    73
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be two non-zero vectors perpendicular to each other and ∣a→∣=∣b→∣|\overrightarrow{a}| = |\overrightarrow{b}|. If ∣a→×b→∣=∣a→∣|\overrightarrow{a}\times\overrightarrow{b}| = |\overrightarrow{a}|, then the angle between the vectors (a→+b→+(a→×b→))(\overrightarrow{a}+\overrightarrow{b}+ (\overrightarrow{a}\times\overrightarrow{b})) and a→\overrightarrow{a} is equal to:

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  2. Question 2

    27 Jun 2022

    165
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be the vectors along the diagonal of a parallelogram having area 222\sqrt{2}. Let the angle between a→\overrightarrow{a} and b→\overrightarrow{b} be acute. ∣a→∣=1|\overrightarrow{a}| = 1 and ∣a→.b→∣=∣a→×b→∣|\overrightarrow{a}.\overrightarrow{b}| = |\overrightarrow{a}\times\overrightarrow{b}|. If c→=22(a→×b→)−2b→\overrightarrow{c}= 2\sqrt{2}(\overrightarrow{a}\times\overrightarrow{b}) - 2\overrightarrow{b}, then an angle between b→\overrightarrow{b} and c→\overrightarrow{c} is:

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  3. Question 3

    29 Jul 2022

    74
    Let aa and bb be two- unit vectors such that the angle between them is π4\frac{\pi}{4}. If θ\theta is the angle between the vectors (a^+b^)(\widehat{a}+\widehat{b}) and (a^+2b^+2(a^×b^))(\widehat{a}+ 2\widehat{b}+ 2(\widehat{a}\times\widehat{b})) then the value of 164cos⁡2θ164\cos^{2}\theta is equal to :

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  4. Question 4

    30 Jan 2023

    159
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be two vectors, Let ∣a→∣=1,∣b→∣=4|\overrightarrow{a}| = 1,|\overrightarrow{b}| = 4 and a→⋅b→=2\overrightarrow{a}\cdot\overrightarrow{b}= 2. If c→=(2a→×b→)−3b→\overrightarrow{c}= (2\overrightarrow{a}\times\overrightarrow{b}) - 3\overrightarrow{b}, then the value of b→⋅c→\overrightarrow{b}\cdot\overrightarrow{c} is

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  5. Question 5

    30 Jan 2024

    158
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be two vectors such that ∣b→∣=1|\overrightarrow{b}| = 1 and ∣b→×a→∣=2|\overrightarrow{b}\times\overrightarrow{a}| = 2. Then ∣(b→×a→)−b→∣2|(\overrightarrow{b}\times\overrightarrow{a}) -\overrightarrow{b}|^{2} is equal to

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  6. Question 6

    30 Jan 2024

    64
    Let a→=a1i^+a2j^+a3k^\overrightarrow{a}=a_{1}\widehat{i}+a_{2}\widehat{j}+a_{3}\widehat{k} and b→=b1i^+b2j^+b3k^\overrightarrow{b}=b_{1}\widehat{i}+b_{2}\widehat{j}+b_{3}\widehat{k} be two vectors such that ∣a→∣=1;a→⋅b→=2|\overrightarrow{a}| = 1;\overrightarrow{a}\cdot\overrightarrow{b}= 2 and ∣b→∣=4|\overrightarrow{b}| = 4. If c→=2(a→×b→)−3b→\overrightarrow{c}= 2(\overrightarrow{a}\times\overrightarrow{b}) - 3\overrightarrow{b}, then the angle between b→\overrightarrow{b} and c→\overrightarrow{c} is equal to:

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  7. Question 7

    31 Jan 2024

    88
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be two vectors such that ∣a→∣=1,∣b→∣=4|\overrightarrow{a}| = 1,|\overrightarrow{b}| = 4 and a→⋅b→=2\overrightarrow{a}\cdot\overrightarrow{b}= 2. If c→=(2a→×b→)−3b→\overrightarrow{c}= (2\overrightarrow{a}\times\overrightarrow{b}) - 3\overrightarrow{b} and the angle between b→\overrightarrow{b} and c→\overrightarrow{c} is α\alpha, then 192sin⁡2α192\sin^{2}\alpha is equal to

  8. Question 8

    7 Apr 2025

    69
    Let the angle θ,0<θ<π2\theta,0 <\theta<\frac{\pi}{2} between two unit vectors a^\widehat{a} and b^\widehat{b} be sin⁡−1(659)\sin^{- 1}\left( \frac{\sqrt{65}}{9} \right). If the vector c→=3a^+6b^+9(a^×b^)\overrightarrow{c}= 3\widehat{a}+ 6\widehat{b}+ 9(\widehat{a}\times\widehat{b}), then the value of 9(c→⋅a^)−3(c→⋅b^)9(\overrightarrow{c}\cdot\widehat{a}) - 3(\overrightarrow{c}\cdot\widehat{b}) is.

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  9. Question 9

    4 Apr 2026 S2

    65
    Let u^\widehat{u} and v^\widehat{v} be unit vectors inclined at an acute angle such that ∣u^×v^∣=32|\widehat{u}\times\widehat{v}| =\frac{\sqrt{3}}{2}. If A→=λu^+v^+(u^×v^)\overrightarrow{A}=\lambda\widehat{u}+\widehat{v}+ (\widehat{u}\times\widehat{v}). Then λ\lambda is equal to :

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