If vec{r} = 3hat{i} + 2hat{j} - 5hat{k},vec{a | vedclass

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

Question

(A) Given $\vec{r} = \lambda\vec{a} + \mu\vec{b} + \gamma\vec{c}$.Substituting the vectors: $3\hat{i} + 2\hat{j} - 5\hat{k} = \lambda(2\hat{i} - \hat{

Vedclass · Accepted Answer

$\mu, \frac{\lambda}{2}, \gamma$ are in $A.P.$

Vedclass · Answer

(A) Given $\vec{r} = \lambda\vec{a} + \mu\vec{b} + \gamma\vec{c}$.Substituting the vectors: $3\hat{i} + 2\hat{j} - 5\hat{k} = \lambda(2\hat{i} - \hat{j} + \hat{k}) + \mu(\hat{i} + 3\hat{j} - 2\hat{k}) + \gamma(-2\hat{i} + \hat{j} - 3\hat{k})$.Equating the coefficients of $\hat{i}, \hat{j}, \hat{k}$:$2\lambda + \mu - 2\gamma = 3$  $(1)$$-\lambda + 3\mu + \gamma = 2$  $(2)$$\lambda - 2\mu - 3\gamma = -5$  $(3)$Solving these equations:From $(2)$,$\gamma = 2 + \lambda - 3\mu$.Substitute $\gamma$ into $(1)$ and $(3)$:$2\lambda + \mu - 2(2 + \lambda - 3\mu) = 3 \implies 7\mu = 7 \implies \mu = 1$.Substitute $\mu = 1$ into $(2)$ and $(3)$:$-\lambda + 3 + \gamma = 2 \implies \gamma - \lambda = -1 \implies \gamma = \lambda - 1$.$\lambda - 2 - 3(\lambda - 1) = -5 \implies -2\lambda + 1 = -5 \implies -2\lambda = -6 \implies \lambda = 3$.Then $\gamma = 3 - 1 = 2$.Thus,$\lambda = 3, \mu = 1, \gamma = 2$.Checking the options: $2\mu = 2(1) = 2$. The sequence $2\mu, \lambda, \gamma$ is $2, 3, 2$ (not $A$.$P$.).Checking $\mu, \frac{\lambda}{2}, \gamma$: $1, 1.5, 2$. This is an $A.P.$ with common difference $0.5$.

If $\vec{r} = 3\hat{i} + 2\hat{j} - 5\hat{k}$,$\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$,$\vec{b} = \hat{i} + 3\hat{j} - 2\hat{k}$,and $\vec{c} = -2\hat{i} + \hat{j} - 3\hat{k}$ such that $\vec{r} = \lambda\vec{a} + \mu\vec{b} + \gamma\vec{c}$,then -

Similar Questions

The points with position vectors $10\,i + 3\,j$,$12\,i - 5\,j$,and $a\,i + 11\,j$ are collinear,if $a = $

Let $a$ and $b$ be unit vectors with $\theta$ as the acute angle between them. If $\frac{1}{2}|a-b|=\sin(\lambda \theta)$,then $4 \lambda^2=$

If two vertices of a triangle are $\hat{i} - \hat{j}$ and $\hat{j} + \hat{k}$,then the third vertex can be

If $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}+3 \hat{j}+\hat{k}$,and $-3 \hat{i}-\hat{j}-2 \hat{k}$ are the position vectors of three points $A$,$B$,and $C$ respectively,then $A$,$B$,and $C$

Represent graphically a displacement of $40 \, km$,$30^{\circ}$ east of north.

Vedclass Test Series

Exam Paper Generator

Online Exam Module