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

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

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.

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.

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

Find a unit vector parallel to the sum of vectors \(\overset\wedge i+\overset\wedge j+\overset\wedge k\)and \(2\overset\wedge i-3\overset\wedge j+5\overset\wedge k\).

Find the scalar components of the vector \(\overset\rightarrow {AB} \) with initial point A(2, 1) and terminal point B(-5, 7).

Write the value of \((\overset\wedge k\times \overset\wedge j).\overset\wedge k i+\overset\wedge i+\overset\wedge j.\overset\wedge k\)

Write the value of \(\left( \hat { k } \times \hat { i } \right) .\hat { i } +\hat { j } .\hat { k } \)

In a triangle ABC, Show that \(\overset\rightarrow {AB}+\overset\rightarrow {BC}+\overset\rightarrow {CA}=0\)

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 projection of \(\overset\rightarrow a+\overset\rightarrow b\) on \(\overset\rightarrow a-\overset\rightarrow b,\)\(\overset\rightarrow a=i+2j+k,\overset\rightarrow b=3\overset\wedge i+\overset\wedge j-\overset\wedge k\).

Given that \(\overrightarrow a.\overrightarrow b=0\) and \(\overrightarrow a\times\overrightarrow b=0\), what can you conclude about the vector \(\overrightarrow a \) and \(\overrightarrow b\)?

Find \(\left| \overrightarrow { a } \right| \) and \(\left| \overrightarrow { b} \right| \) if \((\overset\rightarrow a+\overset\rightarrow b).(\overset\rightarrow a-\overset\rightarrow b)=8\) and \(\left| \overrightarrow { a} \right| =8\left| \overrightarrow { b } \right| .\)

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

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.

Area of rectangle having vertics A, B, C and DWith position vectors: \(\hat{-} i+\frac{1}{2} \hat{j}+4 \hat{k}, \quad \hat{i}+\frac{1}{2} \hat{j}+\hat{4 k}\) \(\hat{i}-\frac{1}{2} \hat{j}+4 \hat{k} \ \text { and }-\hat{i}-\frac{1}{2} \hat{j}+\hat{4 k}\) Respectively, is:
(A) \(\frac { 1 }{ 2 } \)   
(B) 1
(C) 2    
(D) 4

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.

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.

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