Let c be the projection vector of b =λ i +4 k , λ 0 , on the vector a = i +2 j +2 k . If | a + c |=7 , then the area of the parallelogram formed by the vectors b and c is ________

Correct answer: 0 c =( b · a | b | ) a | a | aligned & =((λ+8)/(9))( i +2 j +2 k ) & | a + c |=7 & . ((λ+8)/(9)+1) i +((2(λ+8))/(9)+2) j +((2(λ+8))/(9)+2) k \,=7 & ((λ+8)/(9)+1)^2+((2(λ+8))/(9)+2)^2+((2(λ+8))/(9)+2)^2=49 aligned λ=4 c =(4)/(3) i +(8)/(3) j +(8)/(3) k Area of parallelogram =| array ccc i & j & k (4)/(3) & (8)/(3) & (8)/(3) 4 & 0 & 4 array |=16

JEE Main 202522 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 22 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2025 · 22 January · Morning Shift · Mathematics · Vector Algebra

Let c be the projection vector of b =λ i +4 k , λ 0 , on the vector...

Let

\vec{c}

be the projection vector of

\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda\gt0

, on the vector

\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}

. If

|\vec{a}+\vec{c}|=7

, then the area of the parallelogram formed by the vectors

\vec{b}

and

\vec{c}

is ________

Correct answer

Step-by-step solution

\vec{c}=\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{b}|}\right) \frac{\vec{a}}{|\vec{a}|}

\begin{aligned} & \quad=\left(\frac{\lambda+8}{9}\right)(\hat{i}+2 \hat{j}+2 \hat{k}) \\ & |\vec{a}+\vec{c}|=7 \\ & \left.\Rightarrow\left(\frac{\lambda+8}{9}+1\right) \hat{i}+\left(\frac{2(\lambda+8)}{9}+2\right) \hat{j}+\left(\frac{2(\lambda+8)}{9}+2\right) \hat{k} \right\rvert\,=7 \\ & \left(\frac{\lambda+8}{9}+1\right)^2+\left(\frac{2(\lambda+8)}{9}+2\right)^2+\left(\frac{2(\lambda+8)}{9}+2\right)^2=49\end{aligned}

\Rightarrow \lambda=4 \Rightarrow \vec{c}=\frac{4}{3} \hat{i}+\frac{8}{3} \hat{j}+\frac{8}{3} \hat{k}

Area of parallelogram

=\left|

\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ \frac{4}{3} & \frac{8}{3} & \frac{8}{3} \\ 4 & 0 & 4\end{array}

\right|=16

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