Let a =3 i - j +2 k , ~b = a ×( i -2 k ) and c = b × k . Then the projection of c -2 j on a is :

Correct answer: A. 2 14 aligned & b = a ×( i -3 k ) & =| array ccc i & j & k 3 & -1 & 2 1 & 0 & -2 array |=2 i +8 j + k & c = b × k =8 i -2 j & c -2 j =8 i -4 j aligned Projection of ( i -2 j ) on a aligned & ( c -2 j ) · a = 8,-4,0 · 3,-1,2 14 & = 28 14 =2 14 aligned

JEE Main 202524 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 24 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2025 · 24 January · Evening Shift · Mathematics · Vector Algebra

Let a =3 i - j +2 k , ~b = a ×( i -2 k ) and c = b × k . Then the...

Let

\overrightarrow{\mathrm{a}}=3 \hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\overrightarrow{\mathrm{a}} \times(\hat{i}-2 \hat{k})

and

\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{k}

. Then the projection of

\overrightarrow{\mathrm{c}}-2 \hat{j}

\vec{a}

is :

Options

A $2 \sqrt{14}$
B $\sqrt{14}$
C $3 \sqrt{7}$
D $2 \sqrt{7}$

Correct answer

A. $2 \sqrt{14}$

Step-by-step solution

\begin{aligned} & \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}-3 \hat{\mathrm{k}}) \\ & =\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 3 & -1 & 2 \\ 1 & 0 & -2 \end{array}\right|=2 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{k}}=8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}} \\ & \overrightarrow{\mathrm{c}}-2 \hat{\mathrm{j}}=8 \hat{\mathrm{i}}-4 \hat{\mathrm{j}} \end{aligned}

Projection of

(\hat{\mathrm{i}}-2 \hat{\mathrm{j}})

\overrightarrow{\mathrm{a}}

\begin{aligned} & (\overrightarrow{\mathrm{c}}-2 \hat{\mathrm{j}}) \cdot \hat{\mathrm{a}}=\frac{\langle 8,-4,0\rangle \cdot\langle 3,-1,2\rangle}{\sqrt{14}} \\ & =\frac{28}{\sqrt{14}}=2 \sqrt{14} \end{aligned}

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