Where Y is the Y component of the vector and X is the X component of the vector. “arctan” is the inverse tangent function. This assumes the vector’s tail is located at the origin (0, 0). If the vector is in the second or third quadrant, add 180 degrees to your result. If the vector is in the fourth quadrant, add 360 degrees to your result. Make sure your calculator is set to degrees mode, usually called “deg. ”

Where Y is the Y component of the vector and X is the X component of the vector. “sqrt” is the square root of what’s in the parentheses. This assumes the vector’s tail is located at the origin (0, 0). Check out the full wikiHow guide on finding the magnitude of a vector for more details and examples.

A 2-dimensional plane is a flat surface consisting of two directions, x and y. It extends infinitely in the x and y direction. The x-axis is a horizontal line that measures distance in space in the x direction. Positive values point right from the origin (positive x-axis), negative point left (negative x-axis). The y-axis is a vertical line that measures distance in space in the x direction. It is perpendicular to the x-axis. Positive values point up from the origin (positive y-axis), negative point down (negative y-axis). The origin is where the x and y axes intersect. It has coordinates of (0, 0). Coordinates are where a point is located. They are written as (x, y) where x is the distance you need to move in the horizontal direction and y is the distance you need to move in the vertical direction to get to the point from the origin.

The space above the x-axis and to the right of the y-axis is the first quadrant. Everything in this quadrant is a positive value. It’s the upper right space of the plane. The quadrants are then ordered counterclockwise starting at the first quadrant. So, the second quadrant is above the x-axis and to the left of the y-axis. The third quadrant is below the x-axis and to the left of the y-axis. The fourth quadrant is below the x-axis and to the right of the y-axis.

Vectors are often illustrated as an arrow on a coordinate plane. The tail of the vector is at the origin of the plane. For example, a 2-dimensional vector may have a length of 3 and point 45 degrees counterclockwise from the positive X axis. Note: this wikiHow guide will discuss vectors in a 2-dimensional space, but these principles apply to 3-dimensional spaces as well.

Coordinate notation tells you where the tip of the vector arrow is located. It’s often written as two vertically-stacked numbers within square brackets. The x coordinate is on top while the y is on the bottom. If you don’t know them, you can resolve a vector into components.

u = Xî + Yĵ where u is the vector (this can be any letter, and usually has an arrow pointing right over the letter), X is the x coordinate, and Y is the y coordinate. For example, u = 3î + 4ĵ would be a vector with the tip pointed at the coordinate (3, 4).

Find the angle by starting at the positive x-axis (0 degrees), and then moving counterclockwise until you get to the vector. The vector can point in any direction in the plane, 0 to 360 degrees. This angle will be referred to as 𝛉 (the greek symbol theta).

One side of the triangle will be in line with the x-axis. This side’s length X is the X component of the vector. We’ll refer to this as the adjacent side. The second side is perpendicular to the x-axis. This side’s length Y is the Y component of the vector. We’ll refer to this as the opposite side. The third side of the triangle is the hypotenuse and also the vector itself.

tan(𝛉) = opposite/adjacent where “opposite” is the length of the side farthest from the angle 𝛉 and “adjacent” is the length of the side closest to the angle 𝛉 (that isn’t the hypotenuse)

tan(𝛉) = Y/X

𝛉 = arctan(Y/X) To use your calculator for solving this, first divide Y by X, then press the arctan button. You may need to press the shift or 2nd key on your calculator to access the arctan function. Note: Make sure your calculator is in degrees mode (usually labeled “Deg”). If you’re in radians mode (“Rad”), the answer will be in radians instead of degrees.

The first quadrant doesn’t need an adjustment. For vectors in the second quadrant, the arctan function will result in a negative angle pointing down and to the right (in the fourth quadrant). This angle points directly opposite to the vector, so you’ll need to add 180 degrees to the angle to get 𝛉. For vectors in the third quadrant, the arctan function will result in a positive angle pointing up and to the right (in the first quadrant). This angle points directly opposite to the vector, so you’ll need to add 180 degrees to the angle to get 𝛉. [5] X Research source For vectors in the fourth quadrant, the arctan function will result in a negative angle pointing down and to the right (in the fourth quadrant). This angle points in the correct direction, but the angle is negative (measured clockwise from the positive x-axis) instead of positive (measured counterclockwise from the positive x-axis). You’ll need to add 360 degrees to the angle to get 𝛉.

If one of the vector’s x or y components is 0, then the vector is pointing directly in line with an axis. For example, if you have the vector u = 0î + 5ĵ, the vector is pointing straight up along the positive y-axis. This means it has an angle of 90 degrees since the y-axis is perpendicular to the x-axis. If the vector’s x and y components are the same number, the vector is pointing at an increment of 45 degrees. For example, if you have the vector u = -3î + 3ĵ, the vector is pointing up and to the left, 45 degrees counterclockwise from the positive y-axis. Since the axes are perpendicular, we can add 90 degrees (positive x-axis to positive y-axis) plus 45 degrees to get 135 degrees.

You’re given the vector u = 2î + 5ĵ Insert the components into the tangent equation tan(𝛉) = Y/X tan(𝛉) = 5/2 𝛉 = arctan(5/2) 𝛉 = arctan(2. 5) 𝛉 = 68. 20 degrees

You’re given the vector u = -3î + 6ĵ Insert the components into the tangent equation tan(𝛉) = Y/X tan(𝛉) = 6/-3 𝛉 = arctan(6/-3) 𝛉 = arctan(-2) 𝛉 = -63. 43 degrees This angle points in the fourth quadrant. Apply an adjustment to get the vector angle. 𝛉 = -63. 43 + 180 𝛉 = 116. 57

You’re given the vector u = -1î + -7ĵ Insert the components into the tangent equation tan(𝛉) = Y/X tan(𝛉) = -7/-1 𝛉 = arctan(-7/-1) 𝛉 = arctan(7) 𝛉 = 81. 87 degrees This angle points in the first quadrant. Apply an adjustment to get the vector angle. 𝛉 = 81. 87 + 180 𝛉 = 261. 87

You’re given the vector u = 12î + -4ĵ Insert the components into the tangent equation tan(𝛉) = Y/X tan(𝛉) = -4/12 𝛉 = arctan(-4/12) 𝛉 = arctan(-0. 33) 𝛉 = -18. 43 degrees This angle points in the fourth quadrant, but is negative. Apply an adjustment to get the positive vector angle. 𝛉 = -18. 43 + 360 𝛉 = 341. 57