Calculus help!! PLEASE

Question

Q1) If a and b are two non-parallel, non-zero vectors with |a|=|b|, then why are the vectors u=a+b and v=a−b: 
a) non-zero? 
1) because a and b are non-parallel. 
2) because a and b are non-zero. 
3) because a and b have the same magnitude. 

b) perpendicular (i.e. u⋅v=0)? 
1) because a and b are non-parallel. 
2) because a and b are non-zero. 
3) because a and b have the same magnitude. 



Q2) Suppose there are three points A,B,C with position vectors a=(2,7,−5), b=(−2,3,3) and c=(−14/5,11/5,23/5). 
a) 
(i) Find the vector AB. 

(ii) Find the vector BC. 

b) Are the points A,B and C collinear? 
1) No, since AB≠BC 
2) Yes, since AB=−5BC 
3) Yes, since AB=5BC 
4) No, since |AB|≠|BC| 

Answer 1

Q1).

a).

Because a and b are non parallel vectors, Vectors u = a + b and v = a - b are non zero vectors.

option 1 is the correct choice.

b).

The vectors u = a + b and v = a - b.

Dot product of u and v = u . v

= (a + b) . (a - b)

= a.a + b.a - a.b - b.b

= a^2 - b^2

= a^2 - a^2         (since, |a| = |b|)

= 0.

Therefore, the vectors u and v are perpendecular, because a and b have same  magnitude.

option 3 is the correct choice.

Q2).

a). The points A, B, and C with position vectors a = (2, 7, - 5), b = (- 2, 3, 3), and c = (-14/5, 11/5, 23/5).

(i).

Where,

.

(ii). .

.

b).

If the points A, B, and C are collinear, then the magnitude of AB is equal to magnitude of BC.

Magnitude of vector AB ,.

Magnitude of BC ,.

Since, , they are not collinear.

Option 4 is the correct choice.

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