M L Aggarwal Solutions for Chapter: Vectors, Exercise 4: EXERCISE

Find the value of$\lambda$  so that the vectors  $\hat{i}+3\hat{j},5\hat{k}$ and $\lambda \hat{i}-\hat{j}$ are coplanar.

Find the value of $\lambda$ so that the vectors $\hat{i}-\hat{j} + \hat{k}, 2\hat{i}+ \hat{j}-\hat{k}, \lambda \hat{i}-\hat{j}+\lambda \hat{k}$ are coplanar.

Find the value of $\lambda$ so that the vectors $2\hat{i}-\hat{j}+\hat{k}, \hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda \hat{j}+5\hat{k}$ are coplanar.

If$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$, $\overrightarrow{b}=\hat{i}-\hat{j}+2\hat{k}$, $\overrightarrow{c}=x\hat{i}+(x-2)\hat{j} - \hat{k}$ and the vector $\overrightarrow{c}$ lies in the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$, then find the value of $x$.

If $\overrightarrow{a}= \hat{i}-\hat{k}$ ,$\overrightarrow{b}= x\hat{i}+ \hat{j}+(1- x)\hat{k}$, $\overrightarrow{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$ and, then show that$\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]$ is independent of $x$ and $y$ .

Find the value of $\lambda$ for which the four points $A, B, C$ and $D$ with position vectors $-\hat{j}-\hat{k}, 4\hat{i}+5\hat{j}+\lambda \hat{k}, 3\hat{i}+ 9\hat{j}+4\hat{k}$ and $-4\hat{i}+4\hat{j}+4\hat{k}$ respectively are coplanar.

Find the value of $\lambda$ for which the four points with position vectors $6\hat{i}-7\hat{j}, 16\hat{i}-19\hat{j}-4\hat{k}, \lambda \hat{j}-6\hat{k}$ and $2\hat{i}-5\hat{j}+10\hat{k}$ are coplanar.

Find $\lambda$ such that the four points $A\left(-1, 4, -3\right), B\left(3, \lambda , -5\right), C\left(-3, 8, 5\right)$ and $D\left(-3, 2, 1\right)$ are coplanar.