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Jul 25, 2023 · 011321 Find the equations of the line through parallel to the line with equations. The coefficients of give a direction vector of the given line. Because the line we seek is parallel to this line, also serves as a direction vector for the new line. It passes through , so the parametric equations are.
Sep 8, 2023 · The equation αx +βy +γ = 0 represents a straight line in a 2D coordinate system (a plane). It's in what's called the general form of the equation of a line. To find the direction vector of this line, we can convert this equation into a slope-intercept form, which is more intuitive. The slope-intercept form of a line is y = mx + b, where m is ...
- Overview
- Direction Formula
- Magnitude Formula
- Vector Notation
- Use Trigonometry to Find the Angle
- Special Cases
- Quadrant I Example
- Quadrant II Example
- Quadrant III Example
- Quadrant IV Example
Finding the direction of a vector in a 2-dimensional plane is easy! You’ll just need a little trigonometry. The x and y components of a vector form a right triangle. You can use the tangent function to find the angle between the x-axis and the vector. This wikiHow guide will show you how to find the direction of a vector and walk through four examples. Additionally, we’ll review how to find the magnitude of a vector.
For this method, the vector’s tail will be located at the origin of an xy coordinate plane, and the tip will be at an (X, Y) coordinate.
Use tan (𝛉) = Y/X to find the direction angle 𝛉 of the vector. Apply arctan to both sides to solve for 𝛉.
If your vector is in the second, third, or fourth quadrant, you’ll need to apply an adjustment. Add 180 degrees to your answer for quadrants II and III. Add 360 for quadrant IV.
Use 𝛉 = arctan (Y/X).
This formula calculates 𝛉 (the greek symbol, theta), which is the angle measuring the degrees between the positive x-axis and the vector.
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
Use ||a|| = sqrt (X^2 + Y^2).
This formula calculates ||a||, the magnitude of vector a.
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).
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There are a few coordinate plane terms you’ll need to know.
A Cartesian 2-dimensional coordinate plane specifies points in a plane by assigning distances from the origin.
is a flat surface consisting of two directions, x and y. It extends infinitely in the x and y direction.
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).
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).
is where the x and y axes intersect. It has coordinates of (0, 0).
Locate the angle 𝛉 you’re trying to find.
The direction of a vector can be defined as the angle between the positive x-axis and the vector.
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).
Identify the triangle created by the vector.
There are a few cases where you don’t need to use trigonometry to find the angle of the vector.
These are cases in which the angle of the vector is apparent by looking at it on the graph.
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.
x and y components are the same number
Here’s a step-by-step example of a vector in the first quadrant.
You’re given the vector u = 2î + 5ĵ
Here’s a step-by-step example of a vector in the second quadrant.
You’re given the vector u = -3î + 6ĵ
Insert the components into the tangent equation tan (𝛉) = Y/X
This angle points in the fourth quadrant. Apply an adjustment to get the vector angle.
Here’s a step-by-step example of a vector in the third quadrant.
You’re given the vector u = -1î + -7ĵ
Insert the components into the tangent equation tan (𝛉) = Y/X
This angle points in the first quadrant. Apply an adjustment to get the vector angle.
Here’s a step-by-step example of a vector in the fourth quadrant.
You’re given the vector u = 12î + -4ĵ
Insert the components into the tangent equation tan (𝛉) = Y/X
This angle points in the fourth quadrant, but is negative. Apply an adjustment to get the positive vector angle.
𝛉 = -18.43 + 360
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A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit. Two illustrations of the second convention (discussed above) for identifying the ...
The position vector, ⃑ 𝑟, of any point on a line containing the point 𝑃 (𝑥, 𝑦) with position vector ⃑ 𝑟 is given by ⃑ 𝑟 = ⃑ 𝑟 + 𝑡 ⃑ 𝑑, where ⃑ 𝑑 is the direction vector of the line and 𝑡 is any scalar value. This is called the vector equation of the line.
The vector a is broken up into the two vectors a x and a y (We see later how to do this.) Adding Vectors. We can then add vectors by adding the x parts and adding the y parts: The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)
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The direction of the vector is given by the formula, θ = tan -1 |5/2|. = 68.2° [Because (2, 5) lies in the first quadrant] The direction of the vector is given by 68.2°. Example 2: Consider the image given below. The vector in the above image makes an angle of 50° in the counterclockwise direction with the east.
