Find the magnitude and direction angle for the vector v=3 cos 123 degrees i + 3 sin 123 degrees j.
umhh, the magnitude is 3 and the direction angle is 123°
To find the magnitude and direction angle for the given vector v=3cos(123 degrees)i + 3sin(123 degrees)j, we can use the following formulas:
Magnitude of the vector v:
Magnitude = √(x^2 + y^2)
Direction angle of the vector v:
Direction angle = arctan(y / x)
Let's calculate the values step-by-step:
Step 1: Convert the given vector from polar form to rectangular form.
Using the trigonometric identities:
cos(θ) = x / r
sin(θ) = y / r
Given values:
r = 3 (magnitude of the vector)
θ = 123 degrees
sin(123 degrees) = y / 3
cos(123 degrees) = x / 3
Let's solve for x and y:
y = 3 * sin(123 degrees)
x = 3 * cos(123 degrees)
y ≈ 2.5981
x ≈ -1.3038
Step 2: Calculate the magnitude of the vector v.
Magnitude = √(x^2 + y^2)
Magnitude = √((-1.3038)^2 + (2.5981)^2)
Magnitude ≈ √(1.7045 + 6.7385)
Magnitude ≈ √8.443
Magnitude ≈ 2.9011
Step 3: Calculate the direction angle of the vector v.
Direction angle = arctan(y / x)
Direction angle = arctan(2.5981 / -1.3038)
Direction angle = arctan(-1.9913)
Direction angle ≈ -1.1651 radians
Therefore, the magnitude of the vector v is approximately 2.9011, and the direction angle is approximately -1.1651 radians (or approximately -66.803 degrees).
To find the magnitude and direction angle of the vector v=3 cos 123 degrees i + 3 sin 123 degrees j, we can use the formulas:
Magnitude (or length) of a vector = √(x^2 + y^2)
Direction angle θ = arctan(y / x)
In this case, we have v = 3 cos 123 degrees i + 3 sin 123 degrees j, which means the x-component is 3 cos 123 degrees, and the y-component is 3 sin 123 degrees.
1. Magnitude (or length) of the vector:
Using the formula, we substitute the values:
Magnitude = √((3 cos 123 degrees)^2 + (3 sin 123 degrees)^2)
= √(9 cos^2 (123 degrees) + 9 sin^2 (123 degrees))
= √(9 (cos^2 (123 degrees) + sin^2 (123 degrees)))
= √(9)
= 3
So, the magnitude of the vector v is 3.
2. Direction angle θ:
Using the formula, we substitute the values:
θ = arctan((3 sin 123 degrees) / (3 cos 123 degrees))
θ = arctan(sin 123 degrees / cos 123 degrees)
To calculate the value of θ, we can use a scientific calculator or a computer:
θ ≈ -56.57 degrees
The direction angle θ is approximately -56.57 degrees.
Therefore, the magnitude of the vector v is 3 and the direction angle is approximately -56.57 degrees.