Find the angle between the vectors \(\overrightarrow { a } =\overrightarrow { i } -\overrightarrow { j } +\overrightarrow { k } \ and\ \overrightarrow { b } =\overrightarrow { i } +\overrightarrow { j } -\overrightarrow { k } \)

Find a vector in the direction of \(\overrightarrow { a } =\overrightarrow { i } -2\overrightarrow { j } \) whose magnitude is 7.

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

If \(\overrightarrow { a } =x\hat { i } +2\hat { j } -z\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -y\hat { j } +\hat { k } \) are two equal vectors then write the value of x+y+z.

If \(\left| \overrightarrow { a } \right| =3\) interpret the following:
(i)2\(\overrightarrow { a } \)   
(ii)-5\(\overrightarrow { a } \)

Represent graphically a displacement of 25 km, 60^o .
(i) North to East
(ii) East to North

Find the unit vector in the direction of \(\overset\rightarrow a+\overset\rightarrow b\)if \(\overset\rightarrow a= 2\overset\wedge i+\overset\wedge j+3\overset\wedge k\), and \(\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k\)

Find the direction cosines of the vector joining the points A(1, 2, - 3) and B(- 1, - 2, 1) directed from B to A.

Find \(\left| \overrightarrow {a } \times \overrightarrow { b } \right| \) if \(\left| \overrightarrow { a} \right| =10,\left| \overrightarrow { b } \right| =2\) and \(\overrightarrow a.\overrightarrow b=12\).

Prove that \(\left| \overrightarrow { a } \times \overrightarrow { b } \right| ^{ 2 }=\begin{vmatrix} \overrightarrow { a } .\overrightarrow { a } & \overrightarrow { a } .\overrightarrow { b } \\ \overrightarrow { a } .\overrightarrow { b } & \overrightarrow { b } .\overrightarrow { b } \end{vmatrix}\)

Answer the following as true or false:
(i) \(\overset { \rightarrow }{ a } \) and - \(\overset { \rightarrow }{ a } \)  are collinear.
(ii) The collinear vector are always equal in magnitude
(iii) Two vectors have same magnitude are colinear.
(iv) the collinear vectors having the same magnitude are equal

Find the angle between two vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) with magnitude \(\sqrt { 3 } \) and 2 respectively having \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \) = \(\sqrt { 6 } \).

The value of  \(\hat{i} \cdot(\hat{i} * \hat{k})+\hat{j} \cdot(\hat{i} * \hat{k})+\hat{k} \cdot(\hat{i} * \hat{j}\hat{i} \cdot(\hat{i} * \hat{k})+\hat{j} \cdot(\hat{i} * \hat{k})+\hat{k} \cdot(\hat{i} * \hat{j})\) is:
(A) 0          
(B) -1
(C) 1          
(D)  3.

Find the angle between two vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \)  with magnitudes 1 and 2 respectively and when \(\overset { \rightarrow }{ a } \).\(\overset { \rightarrow }{ b } \) = 1.

Find a vector of magnitude 11 in the direction opposite to that \(\overset { \rightarrow }{ PQ } \) of where P and Q are the points (1,3,2) and (-1,0,8 respectively.

Find the position vector of a point R which divides the line joining teo points P and Q whose position vectors are (\(2\overrightarrow { a } +\overrightarrow { b } \)) and (\(\overrightarrow { a } -3\overrightarrow { b } \)) respectively, externally in the ratio 1:2 Also,show that P is the mid point of the line segment RQ.

If \(\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } \) be the vectors such that \(\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } =\overrightarrow { 0 } ,\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =4\)\( \ and\ \left| \overrightarrow { c } \right| =5\)  find \(\overrightarrow { a } .\overrightarrow { b } +\overrightarrow { b } .\overrightarrow { c } +\overrightarrow { c } .\overrightarrow { a } \)

If \(\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } \)  are unit vectors. Such that \(\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } =\overrightarrow { 0 } \) then find the value of \(\overrightarrow { a } .\overrightarrow { b } +\overrightarrow { b } .\overrightarrow { c } +\overrightarrow { c } .\overrightarrow { a } \)

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 \(-\overset\wedge i+\overset\wedge j-\overset\wedge k\) 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 \(\overrightarrow c\).