Consider the vectors Which pairs if any of these vectors are

Consider the vectors Which pairs (if any) of these vectors are Are perpendicular? (Enter none or a pair or list of pairs, e.g., if a is perpendicular to b and c, enter (a,b),(a,c).) Have an angles less than nl2 between them? Have an angle of more than nil between them?

Solution

a = 4i +j -k ; b = -4i -j +k ; c= i +4j +5k

Dot product : Let a = a1i +a2j +a3k ; b = b1i +b2j +b3k

Let a.b = a1.b1 +a2.b2 +a3b3

Firts of all we can see vector a = -b. S, if two vectors are parallel , they can be exprsessed as a = kb where k is scalar.

So, a and b are parallel

Question b : a and b

Now dot product of b.c = -4*1 -4*1 +5*1 = -3

dot product a.c = 4*1 + 4*1 -5*1 = 3

So in no case dot product is zero, it means none of the vectors are parallel.

Question a : none

To find angle between two vectors : we use cosx = (a.b)/(|a|*|b|) where x in the angle between the two vectors

Angle between a and b : x= o as aand be are parallel

Angle between b and c : b.c = -3 ; |b| = sqrt( 16 +1+1) = sqrt18 = 3sqrt2

cosx = -3/3sqrt2

cosx is -ve it means angle does not lie in Ist quadrant. So, angle between and b is greater than Pi/2

Question d : b and c

So, remaining angle between a and c must be less than pi/2

Question c : a and c