Oh, let's venture into the realm of vectors in mathematics!
For a vector with vx = 3 and vy = 4, we can determine its length and direction using some mathematical magic!
a. To figure it out, we can use the Pythagorean theorem. The length (magnitude) of the vector can be calculated as the square root of the sum of the squares of its components. In this case:
Length = √(vx^2 + vy^2)
= √(3^2 + 4^2)
= √(9 + 16)
= √25
= 5
So, the length of this vector is 5 units.
b. To determine the direction of the vector, we can use trigonometry. We can find the angle θ that the vector makes with the positive x-axis using the formula:
θ = arctan(vy / vx)
θ = arctan(4 / 3)
Using a calculator, we find that θ ≈ 53.13 degrees.
Therefore, the vector with vx = 3 and vy = 4 has a length of 5 units and makes an angle of approximately 53.13 degrees with the positive x-axis.
Now you're ready to navigate the world of vectors, my dear!