Question

Let a and b be arbitrary vectors of constant length and angle, which are neither parallel nor perpendicular, and let c be the vector defined by the equation c=(cost)a + (sint)b. t is greater than or equal to 0.
a) For what values of t is c parallel to a?
b) When is c parallel to b?
c) Can c ever be parallel to a + b?
d) Can c ever be perpendicular to a + b?

Answer 1

c will be parallel to a if it is c= M a where M is some number
in other words, you don't want to add any of vector b.
that means you want sin t = 0 (to make b "go away")

what angle t has sin t = 0 ?

Answer 2

Is it 0 or 180?

Answer 3

yes, or in general n*180º where n is an integer
or n* pi (in radians)

Answer 4

a) For what values of t is c parallel to a? for t= n pi radians, n an integer

Answer 5

So would the value possibly be pi?

Answer 6

yes, pi works. sin(pi)=0
but sin(2pi= 0
sin(3pi)= 0
sin(-6 pi) = 0
and so on

Answer 7

for
b) When is c parallel to b?

it's the same idea, except now you want cos t = 0

Answer 8

So then you would want answers like 2pi/3 or 3pi/2?

Answer 9

pi/2 or 3pi/2 (not 2 pi /3 )
you can test those in your calculator.

Answer 10

Okay. Perfect. All of this makes so much more sense now. For questions c and d the answers would be no then because one of then has to be 0 in order for one of them to be parallel. Or is zero a possible answer or 180 as it was brought up earlier?

Answer 11

c) Can c ever be parallel to a + b?

to be parallel to a vector you have to be a scalar multiple of that vector
for example, for c to be parallel to vector a we need c = M a
where M is a number.

if we can find an M so that
c= M (a + b)
then c will be parallel to (a+b)

in other words, can we find c= M a + M b ?
and that means can we find cos t = sin t (equal to the same number ?)

Answer 12

start with the requirement
sin t = cos t
divide both sides by cos t
\[ \frac{\sin(t)}{\cos(t)} = 1 \]
or
\[ \tan( t) = 1 \]
what angles does tan t =1

Answer 13

Okay. pi/4 and 5pi/4 ?

Answer 14

yes

Answer 15

Perfect! This would be the answer then for c because of the one. Would d be any different?