$\begin{aligned}
& \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \
& \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \
& \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}} \
& \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=0 \
& (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=0 \
& \Rightarrow \overrightarrow{\mathrm{c}}=\lambda(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \
& \overrightarrow{\mathrm{c}}=\lambda(5 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \ldots . .(1) \
& |\overrightarrow{\mathrm{c}}|^2=\lambda^2(25+36+16) \
& |\overrightarrow{\mathrm{c}}|^2=77 \lambda^2 \
& (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}) \cdot(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})=168 \
& \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{~b}}+|\overrightarrow{\mathrm{c}}|^2=168
\end{aligned}14+\vec{c} \cdot(\vec{a}+\vec{b})+77 \lambda^2=168usingequation(1)\begin{aligned}
& \lambda|5 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}|^2+77 \lambda^2=154 \
& 77 \lambda+77 \lambda^2-154=0 \
& \lambda^2+\lambda-2=0 \
& \lambda=-2,1
\end{aligned}\thereforeMaximumvalueof|\overrightarrow{\mathrm{c}}|^2occurswhen\lambda=-2\begin{aligned}
& |\overrightarrow{\mathrm{c}}|^2=77 \lambda^2 \
& =77 \times 4 \
& =308
\end{aligned}$
Let overrightarrow a=2 i- j+3 k, overrightarrow ~b=3 i-5 j+ k and overrightarrow be a vector such that overrightarrow a × overrightarrow…
JEE Main 2025 — Mathematics Vectors & 3D Geometry
Let a=2i^−j^+3k^, b=3i^−5j^+k^ and c be a vector such that a×c=c×b and (a+c)⋅(b+c)=168. Then the maximum value of ∣c∣2 is :
Held on 29 Jan 2025 · Verified 6 Jul 2026.
Options
- A
462
- B
77
- C
154
- D
308
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