Vector Algebra ...

If vectors $$\vec {A} = 2\hat {i} + 3\hat {j} + 4\hat {k}, \vec {B} = \hat {i} + \hat {j} + 5\hat {k}$$ and $$\vec {C}$$ form a left-handed system, then $$\vec {C}$$ is

Let $$\overrightarrow{a}=2\hat{i}-3\hat{j}+6\hat{k}$$ and $$\overrightarrow{b}=-2\hat{i}+3\hat{j}-\hat{k}$$ then projection of $$\overrightarrow{a}$$ on $$\overrightarrow{b} :$$ projection of $$\overrightarrow{b}$$ on $$\overrightarrow{a}=$$

If $$2\overrightarrow{a}+3\overrightarrow{b}+4\overrightarrow{c}=0 \Rightarrow \overrightarrow{a}\times \overrightarrow{b} + \overrightarrow{b}\times \overrightarrow{c} +\overrightarrow{c}\times \overrightarrow{a}=$$

If $$|\vec{a}|=5, |\vec{a}-\vec{b}|=8$$ and $$|\vec{a}+\vec{b}|=10$$, then $$|\vec{b}|$$ is equal to :

Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$$ . A vector in the plane of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, where projection on $$\overrightarrow{c}$$ is $$\dfrac{1}{\sqrt{3}}$$, is

A person travels $$12\ km$$ in the southward direction and then travels $$5\ km$$ to the right and then travels $$15\ km$$ toward the right and finally travels $$5\ km$$ towards the east, how far is he from his starting place?

Let $$\overrightarrow{u}=\hat{i}+\hat{j},\overrightarrow{v}=\hat{i}-\hat{j},\overrightarrow{w}=\hat{i}+2\hat{j}+3\hat{k}$$ If $$\hat{n}$$ is a unit vector such that $$\overrightarrow{u}.\hat{n}=0$$ and $$\overrightarrow {v}.\hat{n}=0$$, then $$\left|\overrightarrow {w}.\hat{n}\right|=$$

Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$$ . A vector in the plane of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, where projection on $$\overrightarrow{c}$$ is $$\dfrac{1}{\sqrt{3}}$$, is

let $$\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$$ be such that $$\left|\overrightarrow{u}\right|=1,\left|\overrightarrow{v}\right|=2,\left|\overrightarrow{w}\right|=3$$. If the projection of $$\overrightarrow{v}$$ along $$\overrightarrow{u}$$ is equal to the projection of $$\overrightarrow{w}$$ along $$\overrightarrow{u}$$ and $$\overrightarrow{v},\overrightarrow{w}$$ are perpendicular to each other, then$$\left|\overrightarrow{u}-\overrightarrow{v}+\overrightarrow{w}\right|=$$

$$D,E\ and \ F$$ are the mid-point of the sides $$BC,CA \ and \ AB$$ respectively of the triangle $$ABC$$. Which of the following is true?