To find the value of [tex]\( c \)[/tex] such that the vector [tex]\(\vec{A} = 0.4 \hat{i} + 0.3 \hat{j} + c \hat{k}\)[/tex] is a unit vector, we need to use the property of unit vectors. A unit vector has a magnitude of 1. Hence, we use the formula for the magnitude of the vector [tex]\(\vec{A}\)[/tex]:
[tex]\[
|\vec{A}| = \sqrt{(0.4)^2 + (0.3)^2 + c^2}
\][/tex]
Since [tex]\(\vec{A}\)[/tex] is a unit vector, its magnitude is 1:
[tex]\[
1 = \sqrt{(0.4)^2 + (0.3)^2 + c^2}
\][/tex]
First, we square both sides to get rid of the square root:
[tex]\[
1 = (0.4)^2 + (0.3)^2 + c^2
\][/tex]
Next, we compute the squares of the given components:
[tex]\[
(0.4)^2 = 0.16
\][/tex]
[tex]\[
(0.3)^2 = 0.09
\][/tex]
So the equation becomes:
[tex]\[
1 = 0.16 + 0.09 + c^2
\][/tex]
Adding 0.16 and 0.09 gives us:
[tex]\[
1 = 0.25 + c^2
\][/tex]
Now, isolate [tex]\( c^2 \)[/tex] by subtracting 0.25 from both sides:
[tex]\[
c^2 = 0.75
\][/tex]
Finally, solve for [tex]\( c \)[/tex] by taking the square root of both sides:
[tex]\[
c = \sqrt{0.75}
\][/tex]
The value of [tex]\(\sqrt{0.75}\)[/tex] can be simplified further. We observe that [tex]\(\sqrt{0.75}\)[/tex] is approximately 0.866.
Thus, the value of [tex]\( c \)[/tex] is approximately 0.866, and the correct choice is:
2) 0.866
