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JEE Main 20245 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Vector Algebra

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Vector Algebra

Let a = i -3 j +7 k , b =2 i - j + k and c be a vector such that (...

Let a→=i^−3j^+7k^,b→=2i^−j^+k^\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k} and c→\overrightarrow{\mathrm{c}} be a vector such that (a→+2b→)×c→=3(c→×a→)(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=3(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}). If a⃗⋅c⃗=130\vec{a} \cdot \vec{c}=130, then b⃗⋅c⃗\vec{b} \cdot \vec{c} is equal to _______

Correct answer

0

Step-by-step solution

(a→+2b→)×c→=3(c→×a→)(2b→+4a→)×c→=0c→=λ(4a→+2b→)=λ(8i^−14j^+30k^)a→⋅c→=1308λ+42λ+210λ=130λ=12c→=4i^−7j^+15k^b→⋅c→=8+7+15=30\begin{aligned} & (\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=3(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}) \\ & (2 \overrightarrow{\mathrm{b}}+4 \overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}=0 \\ & \overrightarrow{\mathrm{c}}=\lambda(4 \overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}})=\lambda(8 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}+30 \hat{\mathrm{k}}) \\ & \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=130 \\ & 8 \lambda+42 \lambda+210 \lambda=130 \\ & \lambda=\frac{1}{2} \\ & \overrightarrow{\mathrm{c}}=4 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}+15 \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=8+7+15=30\end{aligned}
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