R D Sharma Solutions for Chapter: Scalar or Dot Product, Exercise 1: EXERCISE

Find the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $\overrightarrow{a}=3\hat{i}-2\hat{j}-6\hat{k}$ and $\overrightarrow{b}=4\hat{i}-\hat{j}+8\hat{k}$

Find the angles of a triangle the coordinates of whose vertices are $A(0,-1,-2), B(3, 1, 4)$ and $(5, 7, 1)$.

If the vertices $A, B$ and $C$ of $△ABC$ have position vectors $(1, 2, 3), (-1, 0, 0)$and $(0, 1, 2)$, respectively, what is the value of $k$ if the magnitude of $\angle ABC={\cos }^{-1}\left(\frac{10}{\sqrt{k}}\right)$?

Find the projection of $\overrightarrow{b}+\overrightarrow{c}\text{ on }\overrightarrow{a}$ where $\overrightarrow{a}=2\hat{i}-2\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}+2\hat{j}-2\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}$.

If two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are such that $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=1\text{ and }\overrightarrow{a}·\overrightarrow{b}=1$ Then find the value of $(3\overrightarrow{a}-5\overrightarrow{b})·(2\overrightarrow{a}+7\overrightarrow{b})$.

Find the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, if $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=3\text{ and }\overrightarrow{a}·\overrightarrow{b}=1$.

Let $\overrightarrow{a}=5\hat{i}-\hat{j}+7\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\lambda \hat{k}$. Find ${\lambda }^2$ such that $\overrightarrow{a}+\overrightarrow{b}$ is orthogonal to $\overrightarrow{a}-\overrightarrow{b}$.

Let $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$ be vector such $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=\overrightarrow{0}.$ If $\left|\overrightarrow{u}\right|=3,\left|\overrightarrow{v}\right|=4$ and $\left|\overrightarrow{w}\right|=5,$ then find $\overrightarrow{u}·\overrightarrow{v}+\overrightarrow{v}·\overrightarrow{w}+\overrightarrow{w}·\overrightarrow{u}$.