Let a= i + j + k , b= i - j + k and c= i - j - k be three vectors. A vector v in the plane of a and b whose projection of c is 1 3 , is given by

Correct answer: C. 3 i - j +3 k Let v=a+λ b v= (1+λ) i +(1-λ) j +(1+λ) k Projection of v on c= 1 3 array lc & v · c |c| = 1 3 & (1+λ)-(1-λ)-(1+λ) 3 = 1 3 & 1+λ-1+λ-1-λ=1 & λ-1=1 λ=2 & v=3 i - j +3 k array

JEE Advanced 2011Paper 1

Exam: JEE Advanced
Year: 2011
Paper: Paper 1
Subject: Mathematics
Chapter: Vector Algebra

JEE Advanced 2011 · Paper 1 · Mathematics · Vector Algebra

Let a= i + j + k , b= i - j + k and c= i - j - k be three vectors....

Let

\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}

and

\mathbf{c}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}

be three vectors. A vector

\mathbf{v}

in the plane of

\mathbf{a}

and

\mathbf{b}

whose projection of

\mathbf{c}

\frac{1}{\sqrt{3}}

, is given by

Options

A $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
B $-3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$
C $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
D $\hat{\mathbf{i}}+3 \hat{\mathbf{k}}-3 \hat{\mathbf{k}}$

Correct answer

C. $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$

Step-by-step solution

Let

\mathbf{v}=\mathbf{a}+\lambda \mathbf{b} \mathbf{v}=

(1+\lambda) \hat{

\mathbf{i}

}+(1-\lambda) \hat{

\mathbf{j}

}+(1+\lambda) \hat{

\mathbf{k}

} Projection of

\mathbf{v}

\mathbf{c}=\frac{1}{\sqrt{3}}

\begin{array}{lc} \Rightarrow& \frac{\mathbf{v} \cdot \mathbf{c}}{|\mathbf{c}|}=\frac{1}{\sqrt{3}}\\ \Rightarrow & \frac{(1+\lambda)-(1-\lambda)-(1+\lambda)}{\sqrt{3}}=\frac{1}{\sqrt{3}}\\ \Rightarrow & 1+\lambda-1+\lambda-1-\lambda=1 \\ \Rightarrow & \lambda-1=1 \Rightarrow \lambda=2 \\ \therefore & \mathbf{v}=3 \hat{ \mathbf{i} }-\hat{ \mathbf{j} }+3 \hat{ \mathbf{k} } \end{array}

Practice Vector Algebra on Quantrex Academy →

More from Vector Algebra