Scalar and Vector Products

Using vector, find the area of the triangle with vertices  $A(1,1,2),B(2,3,5)\&C(1,5,5).$

Vectors  $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$  are such that  $\left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=\frac{2}{3}\mathrm{and}\left(\overrightarrow{a}\times \overrightarrow{b}\right)$  is a unit vector. Write the angle between  $\overrightarrow{a}\mathrm{and}\overrightarrow{b}.$

If  $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$  are two vectors such that  $\left|\overrightarrow{a}\cdot \overrightarrow{b}\right|=\left|\overrightarrow{a}\times \overrightarrow{b}\right|,$  then what is the angle between  $\overrightarrow{a}\mathrm{and}\overrightarrow{b}?$

The value of p satisfying that $(2\widehat{i}+6\widehat{j}+27\widehat{k})\times (\widehat{i}+3\widehat{j}+p\widehat{k})=0$  would be:

If  $\overrightarrow{p}$  is a unit vector and  $(\overrightarrow{x}-\overrightarrow{p})\cdot (\overrightarrow{x}+\overrightarrow{p})=80,$  then find  $\left|\overrightarrow{x}\right|.$

Choose a unit vector from the given options that is perpendicular to both  $\overrightarrow{a}=2\widehat{i}+\widehat{j}-2\widehat{k}\mathrm{and}\widehat{b}=3\widehat{i}-\widehat{j}+\widehat{k}$ :

If  $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}$  are three mutually perpendicular vectors of equal magnitude, the angle between  $\overrightarrow{a}\mathrm{and}\left(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right)$ would be :

Write the value of  $\left(\widehat{i}\times \widehat{j}\right).\widehat{k}+\widehat{i}.\widehat{j}$

Which of the following is the value of  $\left(\widehat{k}\times \widehat{j}\right).\widehat{i}+\widehat{j}.\widehat{k}.$

What would be the projection of  $\overrightarrow{b}+\overrightarrow{c}\text{on}\overrightarrow{a}\mathrm{where}\overrightarrow{a}=\widehat{i}+2\widehat{j}+\widehat{k},\overrightarrow{b}=\widehat{i}+3\widehat{j}+\widehat{k}\mathrm{on}\overrightarrow{c}=\widehat{i}+\widehat{k}$

The angle between the vectors  $\overrightarrow{a}+\overrightarrow{b}\mathrm{and}\overrightarrow{a}-\overrightarrow{b}$  if  $\overrightarrow{a}=2\widehat{i}-\widehat{j}+3\widehat{k}\mathrm{and}\overrightarrow{b}=3\widehat{i}+\widehat{j}-2\widehat{k}$  is:

The angle between the two vectors  $\overrightarrow{A}=3\widehat{i}+4\widehat{j}+5\widehat{k}\mathrm{and}\overrightarrow{B}=3\widehat{i}+4\widehat{j}-5\widehat{k}$  will be:

The angle between  $\overrightarrow{A}\mathrm{and}\overrightarrow{B}\mathrm{is}\theta .$  The value of the triple product   $\overrightarrow{A}.(\overrightarrow{B}\times \overrightarrow{A})$  is

If  $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A},$  then the angle between  $\overrightarrow{A}$  and  $\overrightarrow{B}$  is –