To find a unit vector in the same direction as a given vector, we first need to determine the magnitude of the vector. The given vector is 8i + j + 4k.
The magnitude (or length) of a vector v = ai + bj + ck is calculated using the formula:
$$ |v| = \sqrt{a^2 + b^2 + c^2} $$
For the vector 8i + j + 4k, the components are:
– a = 8
– b = 1
– c = 4
Substituting these values into the magnitude formula:
$$ |v| = \sqrt{8^2 + 1^2 + 4^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9 $$
Now, to find the unit vector, we divide each component of the vector by its magnitude:
$$ \text{Unit vector} = \frac{1}{|v|}(8i + j + 4k) $$
This gives us:
$$ \text{Unit vector} = \frac{1}{9}(8i + j + 4k) $$
Therefore, the unit vector in the same direction as the given vector 8i + j + 4k is:
\frac{8}{9}i + \frac{1}{9}j + \frac{4}{9}k