Vectors Maths - Vectors

between $\bar{a}$ and $\bar{b}$ is

$(\bar{a}+\bar{b}+\bar{c}) \cdot[(\bar{a}+\bar{b}) \times(\bar{a}+\bar{c})]=-\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$.

$\left[\begin{array}{lll}\bar{a}+\bar{b} & \bar{b}+\bar{c} & \bar{c}+\bar{a}\end{array}\right]=2\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$

vector $\hat{j}+\hat{k} \cdot \hat{i}+\hat{k}$ and $\hat{i}+\hat{j}$. Also find volume of tetrahedron having these

coterminous edges.

$(\bar{a}+\bar{b}+\bar{c}) \times \bar{c}+(\bar{a}+\bar{b}+\bar{c}) \times \bar{b}+(\bar{b}+\bar{c}) \times \bar{a}=2 \bar{a} \times \bar{c}$

Question is modified.

For any vectors $\bar{a}, \bar{b}, \bar{c}$ show that

$(\bar{a}+\bar{b}+\bar{c}) \times \bar{c}+(\bar{a}+\bar{b}+\bar{c}) \times \bar{b}+(\bar{b}-\bar{c}) \times \bar{a}$

$=2 \bar{a} \times \bar{c}$.

$\frac{\pi}{6}$.

Prove that $\hat{a}= \pm 2(\hat{b} \times \hat{c})$

$|\overline{A B} \times \overline{C D}+\overline{B C} \times \overline{A D}+\overline{C A} \times \overline{B D}|=4$ (area of $\triangle A B C$ )

is $\frac{1}{2}[\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}]$

Question is modified.

Show that the area of a triangle $A B C$, the position vectors of whose vertices are $\bar{a}, \bar{b}$ and $\bar{c}$

is $\frac{1}{2}[\bar{a} \times \bar{b}+\bar{b} \times \bar{c}+\bar{c} \times \bar{a}]$.

$\left[\begin{array}{lll}\bar{a} \bar{b} \bar{d}\end{array}\right]+\left[\begin{array}{lll}\bar{b} & \bar{c} & \bar{d}\end{array}\right]+\left[\begin{array}{lll}\bar{c} & \bar{a} & \bar{d}\end{array}\right]=\left[\begin{array}{ll}\bar{a} \bar{b} & \bar{c}\end{array}\right]$

aand $a j+\hat{k}$ becomes minimum.

Question is modified.

Find the value of ' $a$ ' so that the volume of parallelopiped formed by $\hat{i}+a \hat{j}+\hat{k}, \hat{j}+a \hat{k}$

and $a \hat{i}+\hat{k}$ becomes minimum.

vectors in the same plane having projection 1 and 2 along $\bar{b}$ and $\bar{c}_{\text {, respectively, are given Y }}$