Consider the vectors a = i + 5 j - k, b = -i - 5j + k, c = i + j + 6k d = 5i - j

Question

Consider the vectors a = i + 5 j - k, b = -i - 5j + k, c = i + j + 6k d = 5i - j, g = -i - j + 5k. Which pairs (if any) of these vectors are (a) 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).) (b) Are parallel? (Enter none or a pair or list of pairs, e.g., if a is parallel to b and c, enter (a, b), (a, c).) (c) Have an angles less than pi/2 between them? (Enter none or a pair or list of pairs, e.g., if a is at an angle less than pi/2 from b and c, enter (a, b), (a, c).) (d) Have an angle of more than nil between them? (Enter none or a pair or list of pairs, e.g., if a is at an angle greater than pi/2 from b and c, enter (a, b), (a, c).)

Explanation / Answer

Given :

a = i + 5 j - k, b = -i - 5j + k,

c = i + j + 6k, d = 5i - j , g = -i - j + 5k

Which pairs are perpendicular?
The ones whose dot products are zeros:

so, <a.b> = < 1,5,-1> . < -1,-5,1>

= < -1-25-1 >

= -27 that is not equal to 0 , hence they are not perpendicular

now <a.c> = < 1,5,-1> . < 1,1,6>

= < 1+5-6>

=0 , hence <a.c> are perpendicular.

Now, < a.d > = < 1,5,-1> . < 5,-1,0>

= < 5-5-0>

= 0 , Hence ( a.d )are perpendicular.

Now (a.g) = < 1,5,-1>. <-1-1+5>

= < -1 -5 -5 >

= -11 that is not perpendicular.

<b.c> = < -1-5+6>

= 0 , Hence < b.c> are perpendicular.

Hence, ( a.c) , ( a.d), ( b.c) are perpendicular.

#(b) Which pairs are parallel?
(Note that this will include "anti-parallel" pairs.)
The ones that are linear multiples of each other:
(a,b)

#(c)

Which pairs have an angle less than pi/2 between them?
The ones with positive dot-products are (b.g)


#(d) Which pairs have an angle greater than pi/2 between them?
The ones with negative dot-products:
(a,b),(a,g),(b,c),(d,g)

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