Find a unit vector in the direction of (3, -1, 2, 4)
To find a unit vector in the direction of (3, -1, 2, 4), we must first calculate the magnitude of the given vector. The magnitude of a vector (3, -1, 2, 4) is given by the formula:
|v| = sqrt(3^2 + (-1)^2 + 2^2 + 4^2)
|v| = sqrt(9 + 1 + 4 + 16)
|v| = sqrt(30)
To find the unit vector, we divide each component of the given vector by its magnitude:
u = (3/|v|, -1/|v|, 2/|v|, 4/|v|)
u = (3/sqrt(30), -1/sqrt(30), 2/sqrt(30), 4/sqrt(30))
Therefore, a unit vector in the direction of (3, -1, 2, 4) is approximately (0.547, -0.182, 0.365, 0.730).
To find a unit vector in the direction of (3, -1, 2, 4), follow the steps below:
Step 1: Calculate the magnitude of the vector.
- The magnitude of a vector (a, b, c, d) is given by the formula sqrt(a^2 + b^2 + c^2 + d^2).
- For the given vector (3, -1, 2, 4), the magnitude is sqrt(3^2 + (-1)^2 + 2^2 + 4^2) = sqrt(9 + 1 + 4 + 16) = sqrt(30).
Step 2: Divide each component of the vector by its magnitude.
- Dividing each component by the magnitude gives (3/sqrt(30), -1/sqrt(30), 2/sqrt(30), 4/sqrt(30)).
Step 3: Simplify the unit vector.
- We can further simplify the unit vector. Rationalizing the denominator gives (3sqrt(30)/30, -sqrt(30)/30, 2sqrt(30)/30, 4sqrt(30)/30).
- Finally, dividing each term by sqrt(30)/30 gives (3sqrt(30)/sqrt(30), -sqrt(30)/sqrt(30), 2sqrt(30)/sqrt(30), 4sqrt(30)/sqrt(30)).
- This simplifies to the final unit vector (3/5, -1/5, 2/5, 4/5).