Find a unit vector perpendicular to each of the vectors \(\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b } \)  where \(\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k } \)

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

Using vectors find the area of the triangle with vertices A(1,1,2),B(2,3,5) and C(1,5,5).

If \(\overrightarrow { \alpha } =3\hat { i } +4\hat { j } +5\hat { k } \ and\ \overrightarrow { \beta } =2\hat { i } +\hat { j } -4\hat { k } \) the express \(\overrightarrow { \beta } \) in the form \(\overrightarrow { \beta } ={ \overrightarrow { \beta } }_{ 1 }+{ \overrightarrow { \beta } }_{ 2 }\)  where \({ \overrightarrow { \beta } }_{ 1 }\) is parallel to \(\overrightarrow { \alpha } \)  and \({ \overrightarrow { \beta } }_{ 2 }\) is perpendicular to \(\overrightarrow { \alpha } \)

If \(\overrightarrow { a } \ and\ \overrightarrow { b } \) are two vectors such that \(\left| \overrightarrow { a } +\overrightarrow { b } \right| =\left| \overrightarrow { a } \right| \) then prove that vector \(2\overrightarrow { a } +\overrightarrow { b } \) is perpendicular to vector \(\overrightarrow { b } \)

If \(\overrightarrow { a } =\hat { i } -\hat { j } +7\hat { k } \) and \(\overrightarrow { b } =5\hat { i } -\hat { j } +\lambda \hat { k } \) then find the value of \(\lambda\)  so that \(\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b } \) are perpendicular vectors.

Points L,M,N divide the sides BC,CA and AB of a triangle ABC in the ratio 1:4,3:2 and 3:7 respectively. Prove that \(\overrightarrow { AL } +\overrightarrow { BM } +\overrightarrow { CN } \) is a vector parallel to \(\overrightarrow { CK } \) where k divides AB in the ratio 1:3

Find the volume of a parallelopiped whose continuous edges are represented by vectors \(\overrightarrow { a } =2\hat { i } -3\hat { j } +\hat { k } ,\overrightarrow { b } =\hat { i } -\hat { j } +2\hat { k } \) and \(\overrightarrow { c } =2\hat { i } +\hat { j } -\hat { k } \)

Define vector product \(\overrightarrow { a } \times \overrightarrow { b } \) of two vectors \(\overrightarrow { a } \ and\ \overrightarrow { b } \) . If \(\left| \overrightarrow { a } \right| =2,\left| \overrightarrow { b } \right| =5\ and\ \left| \overrightarrow { a } \times \overrightarrow { b } \right| =8\) find the value of \(\overrightarrow { a } .\overrightarrow { b } \)

Mrs. Rodger got a weekly raise of $145. If she gets paid every other week, write an integer describing how the raise will affect her paycheck.

If the vector \(\overrightarrow p=a\overset\wedge i+\overset\wedge j+\overset\wedge k\),\(\overrightarrow q=\overset\wedge i+b\overset\wedge j+\overset\wedge k\) and \(\overrightarrow r=\overset\wedge i+\overset\wedge j+c\overset\wedge k\) are coplannar, then for a, b, c ≠ 1, then show that \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1\)

Prove that : \([\overrightarrow a,\overrightarrow b+\overrightarrow c,\overrightarrow d]=\left[ \begin{matrix} \overrightarrow { a } , & \overrightarrow { b } , & \overrightarrow { d } \end{matrix} \right] +\left[ \begin{matrix} \overrightarrow { a } , & \overrightarrow { c } , & \overrightarrow { d } \end{matrix} \right] \)

If the vector \(-\overset\wedge i+\overset\wedge j-\overset\wedge k\) bisects the angle between the vector \(\overrightarrow c\) and the vector \(3\overset\wedge i+4\overset\wedge j\), then find the vector unit vector in the direction of \(\overrightarrow c\).

If   are unit vectors such that the vector \(\overrightarrow a+3\overrightarrow b\) is perpendicular to \(7\overrightarrow a-5\overrightarrow b\) and \(\overrightarrow a-4\overrightarrow b\) is perpendicular to \(7\overrightarrow a-2\overrightarrow b\), then find the angle between \(\overrightarrow a\) and \(\overrightarrow b\)

If the sum of two unit vectors \(\overrightarrow { a } \) and  \(\overrightarrow { b } \) is a unit vector, then show that the magnitude of their difference is √3.