M L Aggarwal Solutions for Chapter: Vectors, Exercise 2: EXERCISE

If $\left(\overrightarrow{a}-\overrightarrow{b}\right)·\left(\overrightarrow{a}+\overrightarrow{b}\right)=12 \& \left|\overrightarrow{a}\right|=2\left|\overrightarrow{b}\right|$ then find $\left|\overrightarrow{a}\right| +\left|\overrightarrow{b}\right|$. 

If the angle vector $\overrightarrow{a}=3\hat{i}-6\hat{j}+2\hat{k}$ makes with the $x-$axis is $\alpha ={\cos }^{-1}\left(b\right)$ and with $y-$axis is $\beta ={\cos }^{-1}\left(c\right)$ and with $z-$axis is $\gamma ={\cos }^{-1}\left(d\right)$.

Find the value of $b+c+d$.

Dot products of a vector with vectors $3\hat{i}-5\hat{k}, 2\hat{i}+7\hat{j}$ and $\hat{i}+\hat{j}+\hat{k}$ are respectively $-1, 6$ and $5$, if that vector is $x\hat{i}+y\hat{j}+z\hat{k}$, find the value of $x$. 

The dot product of a vector with vectors $\hat{i}+\hat{j}-3\hat{k}, \hat{i}+3\hat{j}-2\hat{k}$ and $2\hat{i}+\hat{j}+4\hat{k}$ are respectively $0, 5$ and $8$, if that vector is $x\hat{i}+y\hat{j}+z\hat{k}$, find the value of $y$. 

If $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\hat{j}+2\hat{k}$, then the angle between $2\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+2\overrightarrow{b}$ is ${\cos }^{-1}\left(\frac{31}{k}\right)$. Find the value of $k$.

Let $\overrightarrow{a}$ and $\overrightarrow{b}$be unit vectors. If the vectors $\overrightarrow{c}=\overrightarrow{a}+2\overrightarrow{b}$ and  $\overrightarrow{d}=5\overrightarrow{a}-4\overrightarrow{b}$ are perpendicular to each other, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\mathrm{\pi }}{k}$, find the value of $k$.
 

If the vectors $\overrightarrow{a}=\hat{i}-\hat{j}+2\hat{k}, \overrightarrow{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}$ are mutually orthogonal, then find the value of $\lambda +\mu$.

Find the value of $\lambda +\mu$ if the vectors $\lambda \hat{i}-3\hat{j}-6\hat{k}$  and $3\hat{i}-\mu \hat{j}-2\hat{k}$ are mutually perpendicular vectors of equal magnitude.