PYQ Vault
EbooksTutor

Split a vector into parts parallel and perpendicular to another vector

This type of question, asked 3 times: 24 Jan 2023, 28 Jan 2025, 4 Apr 2025. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    24 Jan 2023

    154
    Let α→=4i^+3j^+5k^\overrightarrow{\alpha}= 4\widehat{i}+ 3\widehat{j}+ 5\widehat{k} and β→=i^+2j^−4k^\overrightarrow{\beta}=\widehat{i}+ 2\widehat{j}- 4\widehat{k}. Let β→1{\overrightarrow{\beta}}_{1} be parallel to α→\overrightarrow{\alpha} and β→2{\overrightarrow{\beta}}_{2} be perpendicular to α→\overrightarrow{\alpha}. If β→=\overrightarrow{\beta}= β→1+β→2{\overrightarrow{\beta}}_{1}+{\overrightarrow{\beta}}_{2}, then the value of 5β→2⋅(i^+j^+k^)5{\overrightarrow{\beta}}_{2}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) is

    Tap an option to check your answer.

  2. Question 2

    28 Jan 2025

    128
    If the components of a→=αi^+βj^+γk^\overrightarrow{a}= \alpha\widehat{i}+ \beta\widehat{j}+ \gamma\widehat{k} along and perpendicular to b→=3i^+j^−k^\overrightarrow{b}= 3\widehat{i}+\widehat{j}-\widehat{k} respectively, are 1611(3i^+j^−k^)\frac{16}{11}(3\widehat{i}+\widehat{j}-\widehat{k}) and 111(−4i^−5j^−17k^)\frac{1}{11}( - 4\widehat{i}- 5\widehat{j}- 17\widehat{k}), then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is equal to :

    Tap an option to check your answer.

  3. Question 3

    4 Apr 2025

    64
    Consider two vectors u→=3i^−j^\overrightarrow{u}= 3\widehat{i}-\widehat{j} and v→=2i^+j^−λk^,λ>0\overrightarrow{v}= 2\widehat{i}+\widehat{j}-\lambda\widehat{k},\lambda> 0. The angle between them is given by cos⁡−1(527)\cos^{- 1}\left( \frac{\sqrt{5}}{2\sqrt{7}} \right). Let v→=v→1+v→2\overrightarrow{v}={\overrightarrow{v}}_{1}+{\overrightarrow{v}}_{2}, where v→1{\overrightarrow{v}}_{1} is parallel to u→\overrightarrow{u} and v→2{\overrightarrow{v}}_{2} is perpendicular to u→\overrightarrow{u}. Then the value ∣v→1∣2+∣v→2∣2\left| {\overrightarrow{v}}_{1} \right|^{2}+\left| {\overrightarrow{v}}_{2} \right|^{2} is equal to.

    Tap an option to check your answer.