For what value of \(\lambda\) are the vectors \(\overrightarrow { a } =2\overrightarrow { i } +\lambda \overrightarrow { j } +\overrightarrow { k } \) and \(\overrightarrow { b } =\overrightarrow { i } -2\overrightarrow { j } +3\overrightarrow { k } \)  perpendicular to each other?

If \(\overrightarrow { a } =\overrightarrow { i } +2\overrightarrow { j } -\overrightarrow { k } \ and\ \overrightarrow { b } =3\overrightarrow { i } +\overrightarrow { j } -5\overrightarrow { k } \) find a unit vector in the direction of \(\overrightarrow { a } -\overrightarrow { b } \)

Find \(\lambda\), if \((2\hat { i } +6\hat { j } +14\hat { k } )\times (\hat { i } -\lambda \hat { j } +7\hat { k } )=\overrightarrow { 0 } \)

Find the projection of the vector \(\overrightarrow { a } =2\hat { i } +3\hat { j } +2\hat { k } \) on the vector \(\overrightarrow { b } =\hat { i } +2\hat { j } +\hat { k } \)

Write the position vector of the point which divides the join of points with position vectors \(3\overset\rightarrow a-2\overset\rightarrow b\) and \(2\overset\rightarrow a+3\overset\rightarrow b\) in the ratio 2:1

Classify the following on scalar and vector quantities:
(i) Work
(ii) Force
(iii) Velocity
(iv) Displacement.

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 \(\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k\)

If \(\overset\rightarrow a\) and \(\overset\rightarrow b\) are two perpendicular vector then show that \((\overset\rightarrow a+\overset\rightarrow b)^{2}=(\overset\rightarrow a-\overset\rightarrow b)^{2}\)

Find \(\left| \overrightarrow { a} \times \overrightarrow { b } \right| ,\)if \(\overrightarrow a=\overset\wedge i+2\overset\wedge j-\overset\wedge k,\overrightarrow b=3\overset\wedge i+\overset\wedge j-\overset\wedge k\)

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

Find the unit vector in the direction of the vector: \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) 

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

if either vector \(\overset { \rightarrow }{ a } =0\) or \(\overset { \rightarrow }{ b } =0\), then \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =0\) But the converse need not be true. Justify your answer with an example.

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.

Write the direction vector of the vector \(\vec{a}=\hat{i}+\hat{j}-2 \hat{k}\) and hence calculate its direction-consines.

The scalar product of the vector \(\hat { i } +\hat { j } +\hat { k } \) with the unit vector along the sum of vectors \(2\hat { i } +4\hat { j } -5\hat { k } \quad and\quad \lambda \hat { i } +2\hat { j } +3\hat { k } \) is equal to  one. Find the value of \(\lambda\)

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.

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.

Show that the four points A, B, C and D with position vectors \(4\hat { i } +5\hat { j } +\hat { j } ,-\hat { j } -\hat { k } ,3\hat { i } +9\hat { j } +4\hat { k } \) and \(4\left( -\hat { i } +\hat { j } +\hat { k } \right) \) respectively are coplanar.

If \(\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } \) are unit vectors such that \(\overrightarrow { a } .\overrightarrow { b } =\overrightarrow { a } .\overrightarrow { c } =0\) and the angle between \(\overrightarrow { b } \) and \(\overrightarrow { c } \) is \(\frac{\pi}{6}\) , then prove that:
(i) \(\overrightarrow { a } =\pm 2\left( \overrightarrow { b } \times \overrightarrow { c } \right) \)
(ii) \(\left[ \overrightarrow { a } +\overrightarrow { b } ,\overrightarrow { b } +\overrightarrow { c } ,\overrightarrow { c } +\overrightarrow { a } \right] =\pm 1\)