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

If p(1,5,4) and Q(4,1,-2) find the direction ratios of \(\overrightarrow { PQ } \)

Write the direction cosines of the vector \(-2\hat { i } +\hat { j } -5\hat { k } \)

If \(\overrightarrow { a } \) is a unit vector and \((\overrightarrow { x } -\overrightarrow { a } ).(\overrightarrow { x } +\overrightarrow { a } )=8\ find\ \left| \overrightarrow { x } \right| \)

Find the sum of the vectors \(\vec{a}=\hat{i}-2 \hat{j}+\hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}+5 \hat{k} \text { and } \vec{c}=\hat{i}-6 \hat{j}-7 \hat{k}\)

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

Find the position vector of c which divides the line segment joining A & B whose position vectors are \(3\overset\rightarrow a+\overset\rightarrow b\)and \(\overset\rightarrow a-3\overset\rightarrow b\) internally in the ratio 2:3.

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 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 \(\frac{1}{2}.\)

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

For given vectors \(\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k} \text { and } \vec{b}=-\hat{i}+\hat{j}-\hat{k},\) find the unit vector in the direction of \(\vec{a}+\vec{b}\)

Find the unit vector in the direction of the sum of the vector \(\vec{a}=\hat{2} i+2-5 \hat{k} \text { and } \vec{b}=\hat{2} i+\hat{j}+3 \hat{k}\)  

For any two vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \), we always have \(|\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } |\le |\overset { \rightarrow }{ a } ||\overset { \rightarrow }{ b } |\) (cauchy's schwartz inequality)

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are perpendicular vectors, \(|\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } |=13\) and \(\overset { \rightarrow }{ a } \) = 5. Find the value of  \(\overset { \rightarrow }{ b } \)

Find the volume of parallelopiped whose sides are given by vectors: \(2\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } ,\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } and3\overset { \wedge }{ i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \)

If \(\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } \ and\ \overrightarrow { b } =\hat { j } -\hat { k } \) find a vector \(\overrightarrow { c } \) such that \(\overrightarrow { a } \times \overrightarrow { c } =\overrightarrow { b } \ and\ \overrightarrow { a } .\overrightarrow { c } =3\)

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.

Find the unit vector perpendicular to the plane ABC where the position vectors A, B and C are \(2\overset\wedge i-\overset\wedge j+\overset\wedge k,\overset\wedge i+\overset\wedge j+2\overset\wedge k\) and \(2\overset\wedge i+3\overset\wedge k\)

Find the value of \(\lambda\), if the points with position vectors \(3\overset\wedge i-2\overset\wedge j-\overset\wedge k,2\overset\wedge i+3\overset\wedge j-4\overset\wedge k,-\overset\wedge i+\overset\wedge j+2\overset\wedge k\) and \(4\overset\wedge i+5\overset\wedge j+\lambda \overset\wedge k\) 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) \(\left[ \overrightarrow { a } +\overrightarrow { b } ,\overrightarrow { b } +\overrightarrow { c } ,\overrightarrow { c } +\overrightarrow { a } \right] =\pm 1\)