Find a unit vector in the direction of \(\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k } \)

Write the value of p for which \(\overrightarrow { a } =3\hat { i } +2\hat { j } +9\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +p\hat { j } +3\hat { k } \)  are parallel vectors.

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.

Find the projection of vector \(\overset\wedge i+3\overset\wedge j+7\overset\wedge k\) on the vector \(2\overset\wedge i-3\overset\wedge j+6\overset\wedge k\).

Find \(\overrightarrow a.(\overrightarrow b\times\overrightarrow c)\), if \(\overrightarrow a=2\overset\wedge i+\overset\wedge j+3\overset\wedge k,\overrightarrow b=-\overset\wedge i+2\overset\wedge j+\overset\wedge k \ and \ \overrightarrow c=3\overset\wedge i+\overset\wedge j+2\overset\wedge k\)

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

Find the vector of magnitude of 9 units in the direction of \(\vec{a}-\vec{b} \text { if } \vec{a}=3 \hat{i}-2 \hat{j}+3 \hat{k} \text { and } \vec{b}=\hat{i}-4 \hat{j}-\hat{k}\) 

Find the magnitude of two vectors \(\overrightarrow a\) and \(\overrightarrow b\) of equal magnitude such that the angle between them is a 60^o and their scalar product is \(\frac{1}{2}.\)

Classify the following as scalar and vector quantities:
(i) time period     
(ii) distance
(iii)force             
(iv) velocity
(v) work-done.

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

Find the direction-consines of the vector \(\hat{i}+2 \hat{j}+3 \hat{k}\) .

If  \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ a } = 0\) and \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =0\), then what can be conclided about the vector \(\overset { \rightarrow }{ b } \)?

if \(\overset { \rightarrow }{ a } \ \overset { \rightarrow }{ b } \ \overset { \rightarrow }{ c } \)are mutually perpendicular vectors of equal magnitude, show that \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \) is equally inclined to \(\overset { \rightarrow }{ a } \ \overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) 

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.

If the vertices A, B, C of a \(\Delta ABC\) have position vectors (1, 2, 3), (-1, 0, 0) and (0, 1, 2) respectively, what is the magnitude of \(\angle ABC\)?

Let \(\overrightarrow { a } =\hat { i } +4\hat { j } +2\hat { k } ,\overrightarrow { b } =3\hat { i } -2\hat { j } +7\hat { k } \) and \(\overrightarrow { c } =2\hat { i } -\hat { j } +4\hat { k } ,\) find a vector \(\overrightarrow { d } \) which is perpendicular to both \(\overrightarrow { a } \quad and\quad \overrightarrow { b } \quad and\quad \overrightarrow { c } .\overrightarrow { d } =15\)

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 \(\overset { \rightarrow }{ a } \) is any vector in space, show that : \(\overset { \rightarrow }{ a } =(\overset { \rightarrow }{ a } .\overset { \wedge }{ i } )\overset { \wedge }{ i } +(\overset { \rightarrow }{ a } .\overset { \wedge }{ j } )\overset { \wedge }{ j } +(\overset { \rightarrow }{ a } .\overset { \wedge }{ k } )\overset { \wedge }{ k } \)

If \(\overrightarrow a=3\overset\wedge i-\overset\wedge j\) and \(\overrightarrow b=2\overset\wedge i+\overset\wedge j-3\overset\wedge k\), then express \(\overrightarrow b \) in the form of \(\overrightarrow b=\overrightarrow b_1+\overrightarrow b_2\), where\(\overrightarrow b_1 \parallel \overrightarrow a\) and \(\overrightarrow b_2\) perpendicular to \(\overrightarrow a\).

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) \(\left[ \overrightarrow { a } +\overrightarrow { b } ,\overrightarrow { b } +\overrightarrow { c } ,\overrightarrow { c } +\overrightarrow { a } \right] =\pm 1\)