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Expand |pa + qb|^2 using the dot product to find a length, angle or parameter

This type of question, asked 5 times: 2021 Paper 22, 4 Apr 2024, 22 Jan 2025, 7 Apr 2025, 28 Jan 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 22

    66
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be two vectors such that ∣2a→+3b→∣=∣3a→+b→∣|2\overrightarrow{a}+ 3\overrightarrow{b}| = |3\overrightarrow{a}+\overrightarrow{b}| and the angle between a→\overrightarrow{a} and b→\overrightarrow{b} is 60∘60^{\circ}. If 18a→\frac{1}{8}\overrightarrow{a} is a unit vector, then ∣b→∣|\overrightarrow{b}| is equal to:

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  2. Question 2

    4 Apr 2024

    152
    If λ>0\lambda > 0, let θ\theta be the angle between the vectors a→=i^+λj^−3k^\overrightarrow{a}=\widehat{i}+ \lambda\widehat{j}- 3\widehat{k} and b→=3i^−j^+2k^\overrightarrow{b}= 3\widehat{i}-\widehat{j}+ 2\widehat{k}. If the vectors a→+b→\overrightarrow{a}+\overrightarrow{b} and a→−b→\overrightarrow{a}-\overrightarrow{b} are mutually perpendicular, then the value of (14cos⁡θ)2(14\cos\theta)^{2} is equal to

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  3. Question 3

    22 Jan 2025

    140
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be two - unit vectors such that the angle between them is π3\frac{\pi}{3}. If λa→+2b→\lambda\overrightarrow{a}+ 2\overrightarrow{b} and 3a→−λb→3\overrightarrow{a}- \lambda\overrightarrow{b} are perpendicular to each other, then the number of values of λ\lambda in [−1,3]\lbrack - 1,3\rbrack is :

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  4. Question 4

    7 Apr 2025

    127
    Let a→\overrightarrow{a} and b→\overrightarrow{b} be the vectors of the same magnitude such that ∣a→+b→∣+∣a→−b→∣∣a→+b→∣−∣a→−b→∣=\frac{\left| \overrightarrow{a}+\overrightarrow{b} \right|+\left| \overrightarrow{a}-\overrightarrow{b} \right|}{\left| \overrightarrow{a}+\overrightarrow{b} \right|-\left| \overrightarrow{a}-\overrightarrow{b} \right|}= 2+1\sqrt{2}+ 1. Then ∣a→+b→∣2∣a→∣2\frac{|\overrightarrow{a}+\overrightarrow{b}|^{2}}{|\overrightarrow{a}|^{2}} is:

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  5. Question 5

    28 Jan 2026 S1

    69
    For three-unit vectors a→,b→,c→\overrightarrow{a},\overrightarrow{b},\overrightarrow{c} satisfying ∣a→−b→∣2+∣b→−c→∣2+∣c→−a→∣2=9 and |\overrightarrow{a}-\overrightarrow{b}|^{2}+ |\overrightarrow{b}-\overrightarrow{c}|^{2}+ |\overrightarrow{c}-\overrightarrow{a}|^{2}= 9\text{~and~} ∣2a→+kb→+kc→∣=3|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}| = 3 the positive value of kk is :

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