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1.

If the position vectors of the points P and Q are 2 i + 3j + 4k and 3 i – 2 j – 3 k , find the direction cosines of the vector vector PQ and hence prove that,Cos2α + Cos2β + Cos2γ = 1 

Answer»

Let O be the fixed point

 vector OP = position vector of P = 2 i + 3j + 4k = ( 2 , 3 , 4 ) 

vector OQ = position vector of Q = 3 i – 2 j – 3 k = ( 3 , - 2 , -3)

vector PQ = vector (OQ – OP) = (1 , -5 , -7) 

vector │PQ│= √( 1 + 25 + 49 ) = √75 

Direction cosines of vector PQ are Cosα = 1/√ 75 , Cosβ = -5/√ 75 & cosγ = - 7 /√ 75.

 Consider, Cos2α + Cos2β + Cos2γ = 1/75 + 25 /75 + 49 /75 = 1 
Discussion
2.

Using Vectors ,prove that the points ( 2, -1 , 3 ) , ( 3, -5 , 1 ) and ( -1 , 11 , 9 ) are collinear.

Answer»

for all these points to be collinear, they should lie in the same straight line

that is should have same slopes

slope=y2-y1/ x2-x1

slope1 = -5-(-1)/3-2 = -4

slope2= 11-(-5)/ -1-3 = -4

slpoe3= 11-(-1)/ -1-2= -4

 all of them have the same slopes

Therefore they are collinear

thank u
Discussion
3.

Find the angle between the vectors (a + b & a - b) if vector |a| = vector |b|. 

Answer»

Let  vector |a| =  vector |b| = λ ( say)

Consider,  vector (a + b ) • vector ( a - b ) = vector |a|2 - vector |b|2 = λ2 - λ2 = 0. 

Therefore angle between vector (a + b ) & vector (a - b ) is 90°. 
Discussion
4.

If vector (a x b ) x c) = vector (a x ( b x c) , then prove that either vector a is parallel to vector c or vector b is perpendicular to both vector(a , c) 

Answer»

Given that

vector (a x b ) x c) = vector (a x ( b x c ) 

vector (a • c ) b) – vector (b • c ) a) = vector(c • a ) b) – vector( b • a) c)

 Since , vector(a • c) = vector (c • a)

 vector (b • c) a) =  vector (b • a ) c)

 p  vector a = q  vector c

by taking, vector (b • c) = p & vector (b • a)= q 

where , p & q are scalars 

 therefore , vector a is parallel to vector c

Next consider, vector (b • c) a) = vector(b • a) c 

vector (b • c) a) - vector(b • a) c) = 0 

 vector b is perpendicular to both vector (a , c). 
Discussion
5.

The vector −A is: A. greater than Vector A in magnitude B. less than Vector A in magnitude C. in the same direction as Vector A D. in the direction opposite to Vector AE. perpendicular to vector A

Answer»

D. in the direction opposite to Vector A
Discussion
6.

If the magnitude of the sum of two vectors is less than the magnitude of either vector, then: A. the scalar product of the vectors must be negative B. the scalar product of the vectors must be positive C. the vectors must be parallel and in opposite directionsD. the vectors must be parallel and in the same direction E. none of the above

Answer»

A. the scalar product of the vectors must be negative
Discussion
7.

If the magnitude of the sum of two vectors is greater than the magnitude of either vector, then: A. the scalar product of the vectors must be negative B. the scalar product of the vectors must be positive C. the vectors must be parallel and in opposite directions D. the vectors must be parallel and in the same direction E. none of the above

Answer»

E. none of the above 
Discussion
8.

Prove that (i) vector |a + b|2 + vector |a + b|2 = 2 vector { |a |2 + |b |2} (ii) vector |a + b|2 - vector |a + b|2 = 4 vector (a • b)

Answer»

consider, 

vector |a + b|2+ vector |a + b|2 = vector {|a|2 + 2 a • b + | b |2} + vector {| a |2 - 2 a • b + |b|2}

= 2 vector { | a |2 + | b |2 } 

Next consider, 

vector | a + b |2- vector | a + b |2= vector { | a |2 + 2 a • b + | b |2 } - vector { | a |2 - 2 a • b + | b |2 } 

= 4 vector (a • b) 
Discussion
9.

 prove that, ∑  vector (a x ( b x c)) = 0

Answer»

consider,

∑ vector (a x ( b x c)) = vector (a x ( b x c) + vector (b x ( c x a)) + vector (c x ( a x b))

= vector {( c • a ) b) – ( b • a ) c} + vector {(b • a) c) – (b • c) a} + vector {(c • b )a) – ( c • a) b}

= 0
Discussion
10.

Define a null vector ( or Zero vector ) 

Answer»

Solution ; A vector of magnitude zero is called a null vector .
Discussion
11.

Define a scalar and give an example.

Answer»

Solution; Scalar is a physical quantity which has only the magnitude but not the direction. 

Example; mass ,volume , density , speed , etc. 
Discussion
12.

Define a vector and give an example.

Answer»

A vector is a physical quantity, which has both magnitude and direction.

Example; velocity , acceleration , force, etc. 
Discussion
13.

Prove that the vectors 3i –j – 2k & 2i -2j + 4k are orthogonal vectors. 

Answer»

Let vector a = ( 3 , -1 , -2 ) , vector b = ( 2 , -2 , 4) 

Consider vector a • vector b = 6 + 2 – 8 = 0 

Therefore Therefore a and b are orthogonal vectors.  a and Therefore a and b are orthogonal vectors.  b are orthogonal vectors. 
Discussion