for parallel condition of two vectors we know that , a₁ / a₂ = b₁ / b₂ = c₁ / c₂; ( Q = 0 ,cos(0) =1; so ratio will be equal.)

and perpendicular vectors, a₁ a₂ + b₁ b₂ + c₁ c₂ = 0 ( Q = 90 and cos(90) = 0 ) so product of these two vectors is a₁ a₂ + b₁ b₂ + c₁ c₂.

 

1.) a₁ = 1 , b₁ =4, c₁ =5;

a₂ = 3, b₂ = 12, c₂ = 15;

by parallel condition of vector :- a₁ / a₂ = b₁ / b₂ = c₁ / c₂

= 1/3 = 4/12 = 5/15;

1/3 = 1/3 = 1/3;

hence this pair of vector is parallel.

 

2.) a₁ = 4 , b₁ = -8, c₁ = 2;

a₂ = 6, b₂ = 2, c₂ = -4;

by perpendicular condition of vectors, a₁ a₂ + b₁ b₂ + c₁ c₂ = 0

= (4* 6) +(- 8 * 2) + (2 * -4)

=24 - 16 - 8

=0

hence the given pair of vectors are perpendicular to each other.

 

 

3.) These two vectors are neither parallel nor perpendicular.

here,e = 2,6 ,5;

f= 4,12,15;

so ratio will be = 2/4 = 6/12 = 5/15;

1/2 = 1/2   ≠ 1/3;

so these two vectors are not parallel.

for perpendicularity , 2*4 + 6 * 12 + 5*15 ≠ 0;

hence, these two vectors are not perpendicular.

the given pair of vector is neither parallel nor perpendicular