find the magnitude and direction angle of the vector v, v=-2i+5j
please help i have no idea how to do this. could you show me step by step?
Sure, I can help you with that. To find the magnitude and direction angle of a vector, you can follow these steps:
Step 1: Calculate the magnitude (or length) of the vector using the formula:
Magnitude = √(x^2 + y^2)
In this case, the vector is v = -2i + 5j.
So, x = -2 and y = 5.
Magnitude = √((-2)^2 + 5^2)
Magnitude = √(4 + 25)
Magnitude = √29
Therefore, the magnitude of the vector v is √29.
Step 2: Calculate the direction angle (or angle with respect to the positive x-axis) using the formula:
Direction angle (θ) = arctan(y / x)
In this case, y = 5 and x = -2.
Direction angle (θ) = arctan(5 / -2)
Now, we need to find the quadrant in which the vector lies, as arctan function gives an angle within the range [-π/2, π/2].
Since the x component is negative, the vector lies in the second quadrant. Therefore, the direction angle will be greater than 90°.
To get the direction angle in degrees, we need to add 180° to the angle obtained from the inverse tangent function.
Direction angle (θ) = arctan(5 / -2) + 180°
Using a calculator, we can find the angle to be approximately 146.31°.
Therefore, the magnitude of the vector v is √29 and the direction angle is approximately 146.31°.
Of course! I'll guide you step by step on how to find the magnitude and direction angle of the vector v = -2i + 5j.
Step 1: Understand the Components
In this case, we have v = -2i + 5j.
The components of vector v are -2i (the x-component) and 5j (the y-component).
Step 2: Calculate the Magnitude
The magnitude (or length) of a vector can be found using the Pythagorean theorem.
The magnitude of vector v, denoted as |v|, can be calculated as the square root of the sum of the squares of the components:
|v| = √((-2)^2 + 5^2)
So, |v| = √(4 + 25) = √29.
Hence, the magnitude of vector v is √29.
Step 3: Calculate the Direction Angle
The direction angle (or bearing) of a vector is the angle between the vector and the positive x-axis.
To calculate the direction angle, we can use the inverse tangent (or arctan) function, denoted as atan2(y, x), where y is the y-component and x is the x-component of the vector.
In this case, the x-component is -2 and the y-component is 5. So, we have:
Direction angle (θ) = atan2(5, -2)
Using a calculator, find that θ ≈ 110.6 degrees.
Step 4: Determine the Quadrant
The direction angle obtained in Step 3 is the angle measured counterclockwise from the positive x-axis. However, we need to determine the appropriate quadrant for the vector.
Since the x-component is negative (-2i), the vector lies in either the third or the fourth quadrant.
Since the y-component is positive (5j), the vector lies in the fourth quadrant.
Step 5: Finalize the Direction Angle
To finalize the direction angle, we need to add 180 degrees to the angle obtained in Step 3, as the vector lies in the fourth quadrant.
So, the direction angle (θ) ≈ 110.6 + 180 = 290.6 degrees.
Therefore, the magnitude of vector v is √29 and the direction angle is approximately 290.6 degrees.
magnitude : square root of ( (-2)^2 + 5^2 ) = (4 + 25) = sqrt 29 = 5.38516
angle = 180 + tan^-1(5/-2) = 179.9 = 180 degrees.