Let vec{c} and vec{d} be vectors such that |v | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $\vec{c} 	imes (2\hat{i}+3\hat{j}+4\hat{k}) = (2\hat{i}+3\hat{j}+4\hat{k}) 	imes \vec{d}$, which implies $\vec{c} 	imes (2\hat{i}+3\hat{j

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Given $\vec{c} 	imes (2\hat{i}+3\hat{j}+4\hat{k}) = (2\hat{i}+3\hat{j}+4\hat{k}) 	imes \vec{d}$, which implies $\vec{c} 	imes (2\hat{i}+3\hat{j}+4\hat{k}) + \vec{d} 	imes (2\hat{i}+3\hat{j}+4\hat{k}) = 0$. Thus, $(\vec{c}+\vec{d}) 	imes (2\hat{i}+3\hat{j}+4\hat{k}) = 0$. This means $\vec{c}+\vec{d}$ is parallel to $(2\hat{i}+3\hat{j}+4\hat{k})$. Let $\vec{c}+\vec{d} = \lambda(2\hat{i}+3\hat{j}+4\hat{k})$. Given $|\vec{c}+\vec{d}| = \sqrt{29}$, we have $|\lambda| \sqrt{2^2+3^2+4^2} = \sqrt{29}$, so $|\lambda| \sqrt{29} = \sqrt{29}$, which gives $\lambda = \pm 1$. Now, $(\vec{c}+\vec{d}) \cdot (-7\hat{i}+2\hat{j}+3\hat{k}) = \lambda(2\hat{i}+3\hat{j}+4\hat{k}) \cdot (-7\hat{i}+2\hat{j}+3\hat{k}) = \lambda(-14+6+12) = 4\lambda$. For $\lambda = 1$, the value is $4$, and for $\lambda = -1$, the value is $-4$. Thus, $\lambda_1 = 4$ and $\lambda_2 = -4$. The equation becomes $K^2x^2 + (K^2-5K+4)xy + (3K-2)y^2 - 8x + 12y - 4 = 0$. For this to represent a circle, the coefficient of $xy$ must be $0$ and the coefficients of $x^2$ and $y^2$ must be equal. $K^2-5K+4 = 0 \Rightarrow (K-1)(K-4) = 0 \Rightarrow K=1$ or $K=4$. $K^2 = 3K-2 \Rightarrow K^2-3K+2 = 0 \Rightarrow (K-1)(K-2) = 0 \Rightarrow K=1$ or $K=2$. The common value is $K=1$.

Similar Questions

If $a, b, c$ are distinct real numbers and $P, Q, R$ are three points whose position vectors are respectively $a \hat{i}+b \hat{j}+c \hat{k}$, $b \hat{i}+c \hat{j}+a \hat{k}$ and $c \hat{i}+a \hat{j}+b \hat{k}$, then $\angle Q P R=$

If $\overrightarrow{a} \cdot \hat{i}=4$,then $(\overrightarrow{a} \times \hat{j}) \cdot(2 \hat{j}-3 \hat{k})$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0},$ find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}.$

Find the projection of the vector $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ on the vector $\vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module