For v=-5i-2j, find unit vector u in the direction of v, and write your answer as a linear combination of the standard unit vectors i and j.
a. u=-5/7i-2/7j
b. u=-5/29i-2/29j
c. u=-5(sqrt(29))/29i-2(sqrt(29))/29j
d. u=-5(sqrt(7))/7i-2(sqrt(7))/7j
find the length of -5i - 2j
= √(25+4) = √29
divide each component by √29
You might have to rationalize the denominator,
I don't see any radicals in a. and b. , so rule those out
To find the unit vector u in the direction of v, we need to divide v by its magnitude.
First, let's find the magnitude of v. The magnitude of a vector is calculated using the formula: |v| = sqrt(v₁² + v₂²).
Given v = -5i - 2j, we can calculate its magnitude as follows:
|v| = sqrt((-5)² + (-2)²)
= sqrt(25 + 4)
= sqrt(29)
Now, let's divide v by its magnitude to get the unit vector u:
u = (v/|v|)
= (-5/sqrt(29))i - (2/sqrt(29))j
Therefore, the correct answer is c. u = -5(sqrt(29))/29i - 2(sqrt(29))/29j.
To find the unit vector, u, in the direction of vector v, we need to divide v by its magnitude.
First, let's find the magnitude of vector v.
The magnitude of a vector, represented by ||v|| or "norm of v," is calculated using the Pythagorean theorem.
||v|| = √(v_x^2 + v_y^2)
Given v = -5i - 2j, we can plug in the values:
||v|| = √((-5)^2 + (-2)^2)
= √(25 + 4)
= √29
Now, divide vector v by its magnitude to obtain the unit vector, u:
u = v / ||v||
Substituting the values:
u = (-5/√29)i - (2/√29)j
To represent u as a linear combination of the standard unit vectors i and j, we rationalize the denominator (√29) by multiplying the numerator and denominator of each term by √29:
u = (-5/√29)(√29/√29)i - (2/√29)(√29/√29)j
Simplifying:
u = (-5√29/29)i - (2√29/29)j
Therefore, the answer is:
c. u = -5(√29)/29)i - 2(√29)/29)j