M L Aggarwal Solutions for Chapter: Vectors, Exercise 6: CHAPTER TEST

O is the origin and $\overrightarrow{OA}=(1, 2)$ ,$\overrightarrow{OB}=(3, 4)$ and $\overrightarrow{OC}=(x, -2)$. Find by using vector method $x$ so that points $A,B,C$ are collinear.

O is the origin and $\overrightarrow{OA}=(1, 2)$ ,$\overrightarrow{OB}=(3, 4)$ and $\overrightarrow{OC}=(x, -2)$. Find by using vector method $x$ so that points $A,B,C$ form a triangle right-angled at B.

If $\overrightarrow{a}=\hat{i}-\hat{j}$and $\overrightarrow{b}=-\hat{j}+2\hat{k}$ , then find$(\overrightarrow{a}+ \overrightarrow{b})·(\overrightarrow{a}-2\overrightarrow{b})$ .

Find the projection of $\overrightarrow{AB}$ on $\overrightarrow{PQ}$ where $P, Q, A, B$ are the points $(-2,1,3), (0,2,5), (4,-3,0), (7,-5,-1)$ respectively.

If $\left|\overrightarrow{a}+\overrightarrow{b}\right|=60, \left|\overrightarrow{a}-\overrightarrow{b}\right|=40$ and $\left|\overrightarrow{a}\right|=22$ , find$\left|\overrightarrow{b}\right|$ .

The points $A, B, C \text{and} D$ have position vectors $-\hat{i}+\frac{1}{2} \hat{j} +4\hat{k}, \hat{i}+\frac{1}{2}\hat{j}+4\hat{k}, \hat{i}—\frac{1}{2}\hat{j}+4\hat{k}$ and $-\hat{i}-\frac{1}{2}\hat{j}+4\hat{k}$ respectively. If the area of $ABCD$ is $k$ sq. units, then the value of $k$ is

If the volume of the parallelepiped whose coterminous edges are represented by the vectors $5\hat{i}-4\hat{j}+\hat{k}, 4\hat{i}+3\hat{j}+\lambda \hat{k}$ and $\hat{i}-2\hat{j}+7\hat{k}$ is $216$ cubic units, find the value of $\lambda$.

The position vectors of the points $A, B, C$ and $D$ are $3\hat{i}-2\hat{j}-\hat{k}, 2\hat{i}+3\hat{j}-4\hat{k}, -\hat{i}+\hat{j}+2\hat{k}$ and $4\hat{i}+5\hat{j}+\lambda \hat{k}$ respectively. If the points $A, B, C$ and $D$ lie in a plane, then find the value of $\lambda$.