find a unit vector in the direction of u
u = (-5, -12, 9)
a. (-5, -12, 9)
b. 1 / squareroot 26 (-5, -12, 9)
c. squareroot 26 (-5, -12, 9
d. 1/ 5 squareroot 10 (-5, -12, 9)
e. 1/ 5 squareroot 10 (5, 12, -9)
5^2 + 12^2 + 9^2 = 250
length = sqrt 250 = 5 sqrt 10
so
(1 / 5sqrt10) (-5,-12,9)
To find a unit vector in the direction of vector u = (-5, -12, 9), we need to follow these steps:
Step 1: Calculate the magnitude of vector u.
The magnitude of a vector u = (a, b, c) is given by the formula:
|u| = √(a² + b² + c²)
For vector u = (-5, -12, 9), the calculation would be:
|u| = √((-5)² + (-12)² + 9²)
= √(25 + 144 + 81)
= √250
= 5√10
Step 2: Divide each component of vector u by its magnitude to obtain the unit vector.
The unit vector û = (x, y, z) is given by:
û = (u_x / |u|, u_y / |u|, u_z / |u|)
For vector u = (-5, -12, 9), the unit vector can be calculated as follows:
û = (-5 / (5√10), -12 / (5√10), 9 / (5√10))
= (-1 / √10, -12 / √10, 9 / √10)
Comparing the answers given:
a. (-5, -12, 9): This is vector u itself, not the unit vector.
b. 1 / √26 (-5, -12, 9): This is not the correct unit vector.
c. √26 (-5, -12, 9): This is not the correct unit vector.
d. 1 / (5√10) (-5, -12, 9): This is the correct unit vector.
e. 1 / (5√10) (5, 12, -9): This is not the correct unit vector.
Therefore, the correct answer is d. 1 / (5√10) (-5, -12, 9).