Find the magnitude and direction angle of v = -7i + 3j
is this Rachel Morton
sqrt(49+9) = sqrt(58) =7.61
tan angle above -x axis = 3/7
A = .405 radians *180/pi = 23.2 deg
so
angle counterclockwise from x axis = 180-23.2
To find the magnitude and direction angle of vector v = -7i + 3j, we can use trigonometry.
1. Magnitude (|v|):
The magnitude or length of a vector can be calculated using the Pythagorean theorem. For vector v = -7i + 3j, the magnitude is calculated as follows:
|v| = sqrt((-7)^2 + (3)^2)
= sqrt(49 + 9)
= sqrt(58)
≈ 7.62
Therefore, the magnitude of vector v is approximately 7.62.
2. Direction Angle (θ):
The direction angle of a vector is the angle it makes with the positive x-axis.
θ = arctan(y / x)
For vector v = -7i + 3j, the direction angle is calculated as follows:
θ = arctan(3 / -7)
≈ -0.39 radians
To convert the direction angle to degrees, multiply by 180/π:
θ ≈ -0.39 * (180/π)
θ ≈ -22.41 degrees
Therefore, the direction angle of vector v is approximately -22.41 degrees (or -0.39 radians).
To find the magnitude and direction angle of a vector, you can use the following steps:
Step 1: Calculate the magnitude of the vector.
The magnitude (or length) of a vector v = (a, b) can be calculated using the Pythagorean theorem.
The magnitude (|v|) can be found using the formula:
|v| = sqrt(a^2 + b^2)
In this case, the vector v = -7i + 3j, where i and j are the unit vectors in the x and y directions respectively.
Using the formula, we can calculate the magnitude as follows:
|v| = sqrt((-7)^2 + 3^2)
= sqrt(49 + 9)
= sqrt(58)
≈ 7.61577
So, the magnitude of v is approximately 7.61577.
Step 2: Calculate the direction angle of the vector.
The direction angle (θ) of a vector v = (a, b) can be calculated using the formula:
θ = tan^(-1)(b/a)
In this case, the vector v = -7i + 3j.
Using the formula, we can calculate the direction angle as follows:
θ = tan^(-1)(3 / -7)
≈ -22.62°
So, the direction angle of v is approximately -22.62°. Note that this angle is measured counterclockwise from the positive x-axis.
Therefore, the magnitude of v is approximately 7.61577, and the direction angle is approximately -22.62°.