Find All Possible Values of x When 3 Vectors are Linearly Dependent

[fiksx]

That is wrong. can be β² negative?
$\alpha^2+\beta^2=4$ and α=1, what is β²?But what values can β have?No, how come that $1+x^2=\sqrt 3$ ? This must be $\vec c \cdot \vec c$ which is 4.

Can not have β some other value?

If β²=3 β can be √3 or -√3, but √3 is positive. At the end you wrote the correct solution, but take more care to the derivations.

i just wanted to show that $x=\beta$ ,
anyway what i learn from this problem is from a.b=0, dot product=0 show that a and b is orthogonal to each other. so it imply a and b is independent to each other and it span the plane(?) while c is dependent to both a and b, it is in the span of a and b?
but without finding c coordinate, how can you know the angle between b and c can be 150 degree?

 

[ehild]

i just wanted to show that $x=\beta$ ,
anyway what i learn from this problem is from a.b=0, dot product=0 show that a and b is orthogonal to each other. so it imply a and b is independent to each other and it span the plane(?) while c is dependent to both a and b, it is in the span of a and b?
but without finding c coordinate, how can you know the angle between b and c can be 150 degree?

Yes, vector c is in the span of a and b. What is the angle between a and c (both of them)? From that, what can be the angle between b and c?
But I do not see that the angle was asked in the problem.

 

[fiksx]

Yes, vector c is in the span of a and b. What is the angle between a and c (both of them)? From that, what can be the angle between b and c?
But I do not see that the angle was asked in the problem. View attachment 226295

yes from a and c 60 degree, and b and c is 30 degree, i mean there is another way to solve this using, $x=b.c \cos \theta$ ,
$x=$
from $a.c \cos \theta=1$, $1.2 cos \theta=1$, angle between a and c is 60 degree.
$x=b.c \cos \theta$ $x=2.1 cos 30 = \sqrt3$
without finding coordinate, how do you know another angle is 150 degree?

 

[fiksx]

Yes, vector c is in the span of a and b. What is the angle between a and c (both of them)? From that, what can be the angle between b and c?
But I do not see that the angle was asked in the problem. View attachment 226295

ah i think i got it, there is possibility that 60 degree is obtuse or acute angle with b from a

 

[ehild]

yes from a and c 60 degree, and b and c is 30 degree, i mean there is another way to solve this using, $x=b.c \cos \theta$ ,
$x=$
from $a.c \cos \theta=1$, $1.2 cos \theta=1$, angle between a and c is 60 degree.

$1\cdot 2 cos \theta=1$ means that cosθ = 1/2. θ can be 60° or -60°. The angle between b and c can be 30°or 90+60= 150 °. See figure in the previous post.

 

[fiksx]

$1\cdot 2 cos \theta=1$ means that cosθ = 1/2. θ can be 60° or -60°. The angle between b and c can be 30°or 90+60= 150 °. See figure in the previous post.

ok thankyou so much for the help :D learn so much from your answer! sorry I am asking too much !

 

[ehild]

ok thankyou so much for the help :D learn so much from your answer! sorry I am asking too much !

You are welcome