Find a unit vector in the direction of the given vector.
w=-1-2j
Find a unit vector in the direction of the given vector.
w=-i-2j
Sorry I messed up it was not a one it was a i.
w = -i - 2j
describes the vector (-1,-2)
so it the same way I showed you in the similar problem of unit length.
YOu can do it in your head to get (-1/√5 , -2/√5)
To find a unit vector in the direction of the vector w, first, calculate the magnitude of w. The magnitude of a vector can be calculated using the formula:
|w| = √(Re(w)^2 + Im(w)^2)
Given w = -1 - 2j, the real part (Re(w)) is -1 and the imaginary part (Im(w)) is -2.
|w| = √((-1)^2 + (-2)^2)
|w| = √(1 + 4)
|w| = √5
Now, to find the unit vector in the direction of w, divide each component of w by its magnitude:
u = w / |w|
u = (-1 - 2j) / √5
To rationalize the denominator, multiply both the numerator and denominator by the conjugate of √5, which is √5:
u = (-1 - 2j) / √5 × √5 / √5
Simplifying,
u = (-√5 - 2√5j) / 5
Thus, a unit vector in the direction of w is (-√5 - 2√5j) / 5.
To find a unit vector in the direction of a given vector, you need to follow these steps:
1. Calculate the magnitude of the given vector.
2. Divide each component of the vector by its magnitude.
Let's apply these steps to the given vector w = -1 - 2j:
Step 1: Calculate the magnitude of the vector.
The magnitude of a complex number is found by taking the square root of the sum of the squares of its real and imaginary parts. The magnitude (or length) of w can be calculated as follows:
|w| = sqrt((-1)^2 + (-2)^2)
= sqrt(1 + 4)
= sqrt(5)
= 2.236
Step 2: Divide each component of the vector by its magnitude.
To obtain a unit vector (a vector with a magnitude of 1), divide each component of the vector by its magnitude obtained in step 1.
The unit vector u in the direction of w can be calculated as follows:
u = w / |w|
= (-1 - 2j) / 2.236
= -0.447 - 0.894j
Therefore, the unit vector in the direction of the vector w = -1 - 2j is approximately -0.447 - 0.894j.