Write two different having same magnitude.

3

(a)

Find the values of x and y so that the vector \(2 \hat{i}+3 \hat{j} \text { and } x \hat{i}+y \hat{j}\) are equal.

3

(a)

Find out the vector in the direction of vector \(\overset { \rightarrow }{ PQ } \) where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.

3

(a)

Find a vecto in a direction of vector \(5 \hat{i}-\hat{j}+2 \hat{k}\)  which has magnitude 8 units.

3

(a)

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

3

(a)

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

3

(a)

Find \(\overset { \rightarrow }{ |x| } \), if for a unit vector \(\overset { \rightarrow }{ a } ,\left( \overset { \rightarrow }{ x } -\overset { \rightarrow }{ a } \right) .\left( \overset { \rightarrow }{ x } +\overset { \rightarrow }{ a } \right) =12\)

3

(a)

Give that \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =0\) and \(\overset { \rightarrow }{ a } *\overset { \rightarrow }{ b } =0\). What can you conclude about the vector \(\overset { \rightarrow }{ a } \ and \ \overset { \rightarrow }{ b } \).?

3

(a)

If either \(\overset { \rightarrow }{ a } =\overset { \rightarrow }{ 0 } \) or \(\overset { \rightarrow }{ b } =\overset { \rightarrow }{ 0 } \) , then \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =0\) . is the converse true? justify your answer with an example?

3

(a)

If \(\overset { \rightarrow }{ a } =\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \) ,then is it true that \(\overset { \rightarrow }{ |a| } =\overset { \rightarrow }{ |b } |+\overset { \rightarrow }{ |c| } =?\) Justify your answer.

3

(a)

Find the value of x for which \(x(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } )\) is a unit vector.

3

(a)

Prove that \((\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } ).(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } )= \overset { \rightarrow }{ { |a| }^{ 2 } } +\overset { \rightarrow }{ { |b| }^{ 2 } } \) and only if \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \) are perpendicular, given \(\overset { \rightarrow }{ a } \neq 0,\overset { \rightarrow }{ b } \neq 0\) 

3

(a)

Classify the following measure as scalar and vector:
(i) 5 sceonds        
(ii) 1000 cm³
(iii) 10Newton        
(iv) 20 km/hr
(v) 10 g/cm³        
(vi) 20 m/s towards north.

3

(a)

Find the value of x, y and z so that the vector: \(\overset { \rightarrow }{ a } =x\overset { \wedge }{ i } +2\overset { \wedge }{ j } +z\overset { \wedge }{ k } \quad and\quad \overset { \rightarrow }{ b } =2\overset { \wedge }{ i } +y\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are equal.

3

(a)

Show that the points :\(A(2 \hat{i}-\hat{j}+\hat{k}), B(\hat{i}-3 \hat{j}-5 \hat{k}), C(3 \hat{i}-4 \hat{j}-4 \hat{k})\) Are the vertics of a right-angled triangle. 

3

(a)